Erdős · OEIS
← all problemsThe sequences behind the problems.
OEIS · 385 sequences · 461 links
A059442Array of Ramsey numbers R(n,k) (n >= 2, k >= 2) read by antidiagonals.
9 problems2,3,3,4,6,4,5,9,9,5,6,14,18,14,6,7,18,25,25,18,7,8,23
A143824Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.
6 problems0,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10
A000791Ramsey numbers R(3,n).
5 problems1,3,6,9,14,18,23,28,36
A003002Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.
5 problems0,1,2,2,3,4,4,4,4,5,5,6,6,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,14,14,14,14,15,16,16,16,16,16,16,16,
A003003Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.
4 problems1,2,3,3,4,5,5,6,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,17,17,18,18,18,19,20,20,20,21,21,21,22,22,22,23,23,24
A003004Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.
4 problems1,2,3,4,4,5,6,7,8,8,9,10,11,12,12,13,14,15,16,16,16,16,16,17,18,18,19,20,21,21,22,22,23,24,24,25,26,27,28,28,29,30,31,32
A003005Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.
4 problems1,2,3,4,5,5,6,7,8,9,9,10,11,12,13,13,14,15,16,17,17,18,19,20,21,22,22,22,23,23,23,24,25,25,26,27,28,28,29,30,31,31,31,32
A003022Length of shortest (or optimal) Golomb ruler with n marks.
4 problems1,3,6,11,17,25,34,44,55,72,85,106,127,151,177,199,216,246,283,333,356,372,425,480,492,553,585
A227590a(n) = A003022(n)+1 with a(1)=1.
4 problems1,2,4,7,12,18,26,35,45,56,73,86,107,128,152,178,200,217,247,284,334,357,373,426,481,493,554,586
A001223Prime gaps: differences between consecutive primes.
3 problems1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,
A006065Maximal number of 4-tree rows in n-tree orchard problem.
3 problems0,0,0,1,1,1,2,2,3,5,6,7,9,10,12,15,16,18,20,23
A048670Jacobsthal function A048669 applied to the product of the first n primes (A002110).
3 problems2,4,6,10,14,22,26,34,40,46,58,66,74,90,100,106,118,132,152,174,190,200,216,234,258,264,282,300,312,330,354,378,388,414,4
A060355Numbers k such that k and k+1 are powerful numbers.
3 problems8,288,675,9800,12167,235224,332928,465124,1825200,11309768,384199200,592192224,4931691075,5425069447,13051463048,2213222
A186704The minimum number of distinct distances determined by n points in the Euclidean plane.
3 problems0,1,1,2,2,3,3,4,4,5,5,5,6
A000058Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.
2 problems2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,12864938683278671740537145998360961546653259485195807
A005115Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
2 problems2,3,7,23,29,157,907,1669,1879,2089,249037,262897,725663,36850999,173471351,198793279,4827507229,17010526363,83547839407,
A005117Squarefree numbers: numbers that are not divisible by a square greater than 1.
2 problems1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,
A005346Van der Waerden numbers W(2,n).
2 problems1,3,9,35,178,1132
A006037Weird numbers: abundant (A005101) but not pseudoperfect (A005835).
2 problems70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,12110,12530,12670,13370,13510,13790,13930,14770,1561
A006530Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.
2 problems1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,7,11,23,3,5,13,3,7,29,5,31,2,11,17,7,3,37,19,13,5,41,7,43,11,5,23,47,3,7,5,1
A006672a(n) = smallest m such that for every red-blue edge-coloring of the graph K_{m} there exists either a red 4-cycle or a blue K_{1,n}; Ramsey number r(C_4, K_{1,n}).
2 problems4,4,6,7,8,9,11,12,13,14
A006855Maximum number of edges in an n-node squarefree graph, or, in a graph containing no 4-cycle, or no C_4.
2 problems0,1,3,4,6,7,9,11,13,16,18,21,24,27,30,33,36,39,42,46,50,52,56,59,63,67,71,76,80,85,90,92,96,102,106,110,113,117,122,127
A007497a(1) = 2, a(n) = sigma(a(n-1)).
2 problems2,3,4,7,8,15,24,60,168,480,1512,4800,15748,28672,65528,122880,393192,1098240,4124736,15605760,50328576,149873152,3712262
A008997Orchard problem with 5 trees in a row (may not have all been proved optimal).
2 problems0,0,0,0,1,1,1,1,2,2,2,3,3,4,6,6,7,9,10,11
A013928Number of (positive) squarefree numbers < n.
2 problems0,1,2,3,3,4,5,6,6,6,7,8,8,9,10,11,11,12,12,13,13,14,15,16,16,16,17,17,17,18,19,20,20,21,22,23,23,24,25,26,26,27,28,29,29
A014197Number of numbers m with Euler phi(m) = n.
2 problems2,3,0,4,0,4,0,5,0,2,0,6,0,0,0,6,0,4,0,5,0,2,0,10,0,0,0,2,0,2,0,7,0,0,0,8,0,0,0,9,0,4,0,3,0,2,0,11,0,0,0,2,0,2,0,3,0,2,0,
A023193a(n) gives the largest number k for which there is at least one admissible k-tuple taken from [0, 1, ..., n-1] if the tuple starts with 0. Admissibility is defined in a comment.
2 problems1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,12,13,13,14
A039669Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.
2 problems4,7,15,21,45,75,105
A056828Numbers that are not the sum of at most three powerful numbers (A001694).
2 problems7,15,23,87,111,119
A057521Powerful part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n = b*c^2*d^3 then a(n) = c^2*d^3 when b is minimized.
2 problems1,1,1,4,1,1,1,8,9,1,1,4,1,1,1,16,1,9,1,4,1,1,1,8,25,1,27,4,1,1,1,32,1,1,1,36,1,1,1,8,1,1,1,4,9,1,1,16,49,25,1,4,1,27,1,8
A058989Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.
2 problems1,3,5,9,13,21,25,33,39,45,57,65,73,89,99,105,117,131,151,173,189,199,215,233,257,263,281,299,311,329,353,377,387,413,431
A074399a(n) is the largest prime divisor of n(n+1).
2 problems2,3,3,5,5,7,7,3,5,11,11,13,13,7,5,17,17,19,19,7,11,23,23,5,13,13,7,29,29,31,31,11,17,17,7,37,37,19,13,41,41,43,43,11,23,
A076393Decimal expansion of Vardi constant arising in the Sylvester sequence.
2 problems1,2,6,4,0,8,4,7,3,5,3,0,5,3,0,1,1,1,3,0,7,9,5,9,9,5,8,4,1,6,4,6,6,9,4,9,1,1,1,4,5,6,0,1,7,9,2,0,9,0,6,5,5,3,3,1,5,3,4,5,
A098990Decimal expansion of Sum_{n>=1} prime(n)/(2^n).
2 problems3,6,7,4,6,4,3,9,6,6,0,1,1,3,2,8,7,7,8,9,9,5,6,7,6,3,0,9,0,8,4,0,2,9,4,1,1,6,7,7,7,9,7,5,8,8,7,7,9,4,3,7,3,2,8,3,1,2,2,0,
A143301Decimal expansion of the Hall-Montgomery constant.
2 problems1,7,1,5,0,0,4,9,3,1,4,1,5,3,6,0,6,5,8,6,0,4,3,9,9,7,1,5,5,5,2,1,2,1,0,9,6,2,2,2,6,2,9,0,4,2,2,9,5,5,0,8,4,1,7,1,4,2,1,1,
A146968Brocard's problem: positive integers n such that n!+1 = m^2.
2 problems4,5,7
A167485Smallest positive integer m such that n can be expressed as the sum of an initial subsequence of the divisors of m, or 0 if no such m exists.
2 problems1,1,0,2,3,0,5,4,7,15,12,21,6,9,13,8,12,30,10,42,19,18,20,57,14,36,46,30,12,102,29,16,21,42,62,84,22,36,37,18,27,63,20,50
A186705The Erdős unit distance problem: the maximum number of occurrences of the same distance among n points in the plane.
2 problems0,1,3,5,7,9,12,14,18,20,23,27,30,33,37,41,43,46,50,54,57
A263647Numbers k such that 2^k-1 and 3^k-1 are coprime.
2 problems1,2,3,5,7,9,13,14,15,17,19,21,25,26,27,29,31,34,37,38,39,41,45,47,49,51,53,57,59,61,62,63,65,67,71,73,74,79,81,85,87,89,
A264810Number of numbers k <= n such that phi(m) = k for some m.
2 problems1,2,2,3,3,4,4,5,5,6,6,7,7,7,7,8,8,9,9,10,10,11,11,12,12,12,12,13,13,14,14,15,15,15,15,16,16,16,16,17,17,18,18,19,19,20,2
A292528Minimal number of vertices in a triangle-free graph with chromatic number n.
2 problems1,2,5,11,22
A384927a(n) is the maximum size of a subset S of {1, 2, ..., n} such that for any distinct elements t, u in S, t + u does not divide t*u.
2 problems1,2,3,4,5,5,6,7,8,9,10,10,11,12,12,13,14,14,15,15,16,17,18,18,19,20,21,21,22,23,24,25,26,27,27,28,29,30,31,31,32,32,33,3
A391668Table read by antidiagonals. T(n,k) is the least number coprime to all numbers in [n+1, n+k].
2 problems3,5,2,5,5,3,7,7,3,2,7,7,7,7,5,11,11,11,11,5,2,11,11,11,11,5,3,3,11,11,11,11,5,5,5,2,11,11,11,11,11,11,7,7,3,13,13,13,13,
A392164a(n) is the size of the largest subset S of {1,...,N} such that every element of S+S is squarefree.
2 problems1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,
A000001Number of groups of order n.
1 problem0,1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,1,2,1,14,1,2,2,14,1,6,1,4,2,2,1,52,2,5,1,5,1,15,2,1
A000051a(n) = 2^n + 1.
1 problem2,3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,131073,262145,524289,1048577,2097153,4194305,8388609,
A000372Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generat
1 problem2,3,6,20,168,7581,7828354,2414682040998,56130437228687557907788,286386577668298411128469151667598498812366
A000445Latest possible occurrence of the first consecutive pair of n-th power residues, modulo any prime.
1 problem9,77,1224,7888,202124,1649375
A001009Triangle giving number L(n,k) of normalized k X n Latin rectangles.
1 problem1,1,1,1,1,1,1,3,4,4,1,11,46,56,56,1,53,1064,6552,9408,9408,1,309,35792,1293216,11270400,16942080,16942080,1,2119,1673792
A001034Orders of noncyclic simple groups (without repetition).
1 problem60,168,360,504,660,1092,2448,2520,3420,4080,5616,6048,6072,7800,7920,9828,12180,14880,20160,25308,25920,29120,32736,3444
A001220Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.
1 problem1093,3511
A001274Numbers k such that phi(k) = phi(k+1).
1 problem1,3,15,104,164,194,255,495,584,975,2204,2625,2834,3255,3705,5186,5187,10604,11715,13365,18315,22935,25545,32864,38804,39
A001438Maximal number of mutually orthogonal Latin squares (or MOLS) of order n.
1 problem1,2,3,4,1,6,7,8
A001661Largest number not the sum of distinct positive n-th powers.
1 problem128,12758,5134240,67898771,11146309947,766834015734,4968618780985762
A001694Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).
1 problem1,4,8,9,16,25,27,32,36,49,64,72,81,100,108,121,125,128,144,169,196,200,216,225,243,256,288,289,324,343,361,392,400,432,4
A002048Segmented numbers, or prime numbers of measurement.
1 problem1,2,4,5,8,10,14,15,16,21,22,25,26,28,33,34,35,36,38,40,42,46,48,49,50,53,57,60,62,64,65,70,77,80,81,83,85,86,90,91,92,10
A002182Highly composite numbers: numbers n where d(n), the number of divisors of n (A000005), increases to a record.
1 problem1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,7560,10080,15120,20160,25200,27720,45360,50400,55440,
A002202Values taken by totient function phi(m) (A000010).
1 problem1,2,4,6,8,10,12,16,18,20,22,24,28,30,32,36,40,42,44,46,48,52,54,56,58,60,64,66,70,72,78,80,82,84,88,92,96,100,102,104,10
A002233a(1) = 1; for n > 1, a(n) = least positive prime primitive root of n-th prime.
1 problem1,2,2,3,2,2,3,2,5,2,3,2,7,3,5,2,2,2,2,7,5,3,2,3,5,2,5,2,11,3,3,2,3,2,2,7,5,2,5,2,2,2,19,5,2,3,2,3,2,7,3,7,7,11,3,5,2,43,
A002264Nonnegative integers repeated 3 times.
1 problem0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,
A002386Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.
1 problem2,3,7,23,89,113,523,887,1129,1327,9551,15683,19609,31397,155921,360653,370261,492113,1349533,1357201,2010733,4652353,170
A002503Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.
1 problem5,14,27,41,44,65,76,90,109,125,139,152,155,169,186,189,203,208,209,219,227,230,237,265,275,298,307,311,314,321,324,329,3
A002583Largest prime factor of n! + 1.
1 problem2,2,3,7,5,11,103,71,661,269,329891,39916801,2834329,75024347,3790360487,46271341,1059511,1000357,123610951,1713311273363
A002827Unitary perfect numbers: numbers k such that usigma(k) - k = k.
1 problem6,60,90,87360,146361946186458562560000
A002858Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.
1 problem1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102,106,114,126,131,138,145,148,155,175,177,180,
A002975Primitive weird numbers: weird numbers with no proper weird divisors.
1 problem70,836,4030,5830,7192,7912,9272,10792,17272,45356,73616,83312,91388,113072,243892,254012,338572,343876,388076,519712,539
A003015Numbers that occur 5 or more times in Pascal's triangle.
1 problem1,120,210,1540,3003,7140,11628,24310,61218182743304701891431482520
A003016Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).
1 problem0,3,1,2,2,2,3,2,2,2,4,2,2,2,2,4,2,2,2,2,3,4,2,2,2,2,2,2,4,2,2,2,2,2,2,4,4,2,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,2,2,4,4,2,2,2,
A003034Sylvester's problem: minimal number of ordinary lines through n points in the plane.
1 problem3,3,4,3,3,4,6,5,6,6,6,7
A003035Maximal number of 3-tree rows in n-tree orchard problem.
1 problem0,0,1,1,2,4,6,7,10,12,16,19,22,26
A003135n! is a nontrivial product of factorials. It is conjectured that the list is complete.
1 problem9,10,16
A003142Largest subset of 3 X 3 X ... X 3 cube (in n dimensions) with no 3 points collinear.
1 problem0,2,6,16,43,124,353
A003323Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's.
1 problem2,3,6,17
A003458Erdős-Selfridge function: a(n) is the least number m > n+1 such that the least prime factor of binomial(m, n) is > n.
1 problem3,6,7,7,23,62,143,44,159,46,47,174,2239,239,719,241,5849,2098,2099,43196,14871,19574,35423,193049,2105,36287,1119,284,24
A003829Maximal number of unit circles through n points in plane, each circle containing 3 of the points.
1 problem1,4,4,8,12,16
A004059a(n) gives position of first n in A057561.
1 problem1,2,4,5,6,8,9,11,13,14,15,17,18,20,22,23,24,26,28,29,30,32,34,35,36,38,40,41,42,43,45,47,48,50,51,53,55,56,57,59,60,61,6
A004526Nonnegative integers repeated, floor(n/2).
1 problem0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23
A004539Expansion of sqrt(2) in base 2.
1 problem1,0,1,1,0,1,0,1,0,0,0,0,0,1,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,0,
A005113Smallest prime in class n (sometimes written n+) according to the Erdős-Selfridge classification of primes.
1 problem2,13,37,73,1021,2917,15013,49681,532801,1065601,8524807,68198461,545587687,1704961513,23869461181,288310406533,183317462
A005153Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.
1 problem1,2,4,6,8,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,72,78,80,84,88,90,96,100,104,108,112,120,126,128,132,140,14
A005185Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.
1 problem1,1,2,3,3,4,5,5,6,6,6,8,8,8,10,9,10,11,11,12,12,12,12,16,14,14,16,16,16,16,20,17,17,20,21,19,20,22,21,22,23,23,24,24,24,
A005235Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.
1 problem3,5,7,13,23,17,19,23,37,61,67,61,71,47,107,59,61,109,89,103,79,151,197,101,103,233,223,127,223,191,163,229,643,239,157,1
A005236Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n.
1 problem2,3,4,5,6,8,9,10,12,14,17,18,20,24,26,28,30,33,38,42,48,50,54,60,65,74,82,84,90,98,102,108,110,114,126,129,138,150,164,1
A005237Numbers k such that k and k+1 have the same number of divisors.
1 problem2,14,21,26,33,34,38,44,57,75,85,86,93,94,98,104,116,118,122,133,135,141,142,145,147,158,171,177,189,201,202,205,213,214,
A005243A self-generating sequence: start with 1 and 2, take all sums of any number of successive previous elements and adjoin them to the sequence. Repeat!
1 problem1,2,3,5,6,8,10,11,14,16,17,18,19,21,22,24,25,29,30,32,33,34,35,37,40,41,43,45,46,47,49,51,54,57,58,59,60,62,65,67,68,69,
A005244A self-generating sequence: start with 2 and 3, take all products of any 2 previous elements, subtract 1 and adjoin them to the sequence.
1 problem2,3,5,9,14,17,26,27,33,41,44,50,51,53,65,69,77,80,81,84,87,98,99,101,105,122,125,129,131,134,137,149,152,153,158,159,161
A005250Record gaps between primes.
1 problem1,2,4,6,8,14,18,20,22,34,36,44,52,72,86,96,112,114,118,132,148,154,180,210,220,222,234,248,250,282,288,292,320,336,354,3
A005278Noncototients: numbers k such that x - phi(x) = k has no solution.
1 problem10,26,34,50,52,58,86,100,116,122,130,134,146,154,170,172,186,202,206,218,222,232,244,260,266,268,274,290,292,298,310,326
A005279Numbers having divisors d, e with d < e < 2*d.
1 problem6,12,15,18,20,24,28,30,35,36,40,42,45,48,54,56,60,63,66,70,72,75,77,78,80,84,88,90,91,96,99,100,102,104,105,108,110,112,
A005282Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.
1 problem1,2,4,8,13,21,31,45,66,81,97,123,148,182,204,252,290,361,401,475,565,593,662,775,822,916,970,1016,1159,1312,1395,1523,15
A005420Largest prime factor of 2^n - 1.
1 problem3,7,5,31,7,127,17,73,31,89,13,8191,127,151,257,131071,73,524287,41,337,683,178481,241,1801,8191,262657,127,2089,331,2147
A005487Starts 0, 4 and contains no 3-term arithmetic progression.
1 problem0,4,5,7,11,12,16,23,26,31,33,37,38,44,49,56,73,78,80,85,95,99,106,124,128,131,136,143,169,188,197,203,220,221,226,227,23
A006036Primitive pseudoperfect numbers.
1 problem6,20,28,88,104,272,304,350,368,464,490,496,550,572,650,748,770,910,945,1184,1190,1312,1330,1376,1430,1504,1575,1610,1696
A006197Least number not dividing binomial(2n,n).
1 problem3,4,3,3,5,5,5,4,3,3,5,3,3,7,7,4,7,8,9,8,7,7,7,7,5,5,3,3,9,3,3,4,8,8,5,3,3,9,3,3,13,13,13,11,11,11,11,8,7,5,5,5,13,9,5,5,
A006285Odd numbers not of form p + 2^k (de Polignac numbers).
1 problem1,127,149,251,331,337,373,509,599,701,757,809,877,905,907,959,977,997,1019,1087,1199,1207,1211,1243,1259,1271,1477,1529,
A006286Numbers not of form p + 2^x + 2^y.
1 problem1,2,3,128,150,252,332,338,374,510,600,702,758,810,878,906,908,960,978,998,1020,1088,1200,1208,1212,1244,1260,1272,1478,1
A006370The Collatz or 3x+1 map: a(n) = n/2 if n is even, 3n + 1 if n is odd.
1 problem0,4,1,10,2,16,3,22,4,28,5,34,6,40,7,46,8,52,9,58,10,64,11,70,12,76,13,82,14,88,15,94,16,100,17,106,18,112,19,118,20,124,
A006517Numbers k such that k divides 2^k + 2.
1 problem1,2,6,66,946,8646,180246,199606,265826,383846,1234806,3757426,9880278,14304466,23612226,27052806,43091686,63265474,66154
A006521Numbers n such that n divides 2^n + 1.
1 problem1,3,9,27,81,171,243,513,729,1539,2187,3249,4617,6561,9747,13203,13851,19683,29241,39609,41553,59049,61731,87723,97641,11
A006560Smallest starting prime for n consecutive primes in arithmetic progression.
1 problem2,2,3,251,9843019,121174811
A006585Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.
1 problem1,0,1,6,72,2320,245765,151182379
A006856Maximal number of edges in n-node graph of girth at least 5.
1 problem0,1,2,3,5,6,8,10,12,15,16,18,21,23,26,28,31,34,38,41,44,47,50,54,57,61,65,68,72,76,80,85,87,90,95,99,104,109,114,120,124
A006931Least Carmichael number with n prime factors, or 0 if no such number exists.
1 problem561,41041,825265,321197185,5394826801,232250619601,9746347772161,1436697831295441,60977817398996785,7156857700403137441,
A007539a(n) = first n-fold perfect (or n-multiperfect) number.
1 problem1,6,120,30240,14182439040,154345556085770649600,141310897947438348259849402738485523264343544818565120000
A007865Number of sum-free subsets of {1, ..., n}.
1 problem1,2,3,6,9,16,24,42,61,108,151,253,369,607,847,1400,1954,3139,4398,6976,9583,15456,20982,32816,45417,70109,94499,148234,2
A008407Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.
1 problem0,2,6,8,12,16,20,26,30,32,36,42,48,50,56,60,66,70,76,80,84,90,94,100,110,114,120,126,130,136,140,146,152,156,158,162,168
A008908a(n) = (1 + number of halving and tripling steps to reach 1 in the Collatz (3x+1) problem), or -1 if 1 is never reached.
1 problem1,2,8,3,6,9,17,4,20,7,15,10,10,18,18,5,13,21,21,8,8,16,16,11,24,11,112,19,19,19,107,6,27,14,14,22,22,22,35,9,110,9,30,17
A014544Numbers k such that a cube can be divided into k subcubes.
1 problem1,8,15,20,22,27,29,34,36,38,39,41,43,45,46,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73
A015919Positive integers k such that 2^k == 2 (mod k).
1 problem1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157
A015921Positive integers n such that 2^n == 4 (mod n).
1 problem1,2,4,6,10,12,14,22,26,30,34,38,46,58,62,74,82,86,94,106,118,122,132,134,142,146,158,166,170,178,182,194,202,206,214,218
A015940Positive integers n such that 2^n == -3 (mod n).
1 problem1,5,917,3223,62911,326329,395819,33504053,4446226763,17556128765,141613728437,5259417592253,113837290408523
A025418Least sum of 3 positive cubes in exactly n ways.
1 problem3,251,5104,13896,161568,1296378,2016496,2562624,14926248,34012224,69190848,150547032,119095488,1204376256,952763904,1592
A025456Number of partitions of n into 3 positive cubes.
1 problem0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
A025494Squares which are the sum of factorials of distinct integers (probably finite).
1 problem1,4,9,25,121,144,729,841,5041,5184,45369,46225,363609,403225,3674889,1401602635449
A027434a(1) = 2; then defined by property that a(n) = smallest number >= a(n-1) such that successive runs have lengths 1,1,2,2,3,3,4,4.
1 problem2,3,4,4,5,5,6,6,6,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14
A027627Maximal cardinality of 2-distance set in R^n.
1 problem3,5,6,10,16,27,29,45
A028391a(n) = n - floor(sqrt(n)).
1 problem0,0,1,2,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,19,20,20,21,22,23,24,25,26,27,28,29,30,30,31,32,33,34,35,36,37,3
A030126Schur's numbers (version 1).
1 problem2,5,14,45,161
A030659Smallest possible maximum denominator in an expression for 1 as a sum of n distinct unit (Egyptian) fractions.
1 problem6,12,15,15,18,20,24,24,28,30,33,33,35,36,40,42,48,52,52,54,55,55,56,60,63,72,75,75,76,76,77,78,80,85,85,88,90,95,96,96,1
A030979Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.
1 problem0,1,10,756,757,3160,3186,3187,3250,7560,7561,7651,20007,59548377,59548401,45773612811,45775397187,237617431723407,249919
A032742a(1) = 1; for n > 1, a(n) = largest proper divisor of n (that is, for n>1, maximum divisor d of n in range 1 <= d < n).
1 problem1,1,1,2,1,3,1,4,3,5,1,6,1,7,5,8,1,9,1,10,7,11,1,12,5,13,9,14,1,15,1,16,11,17,7,18,1,19,13,20,1,21,1,22,15,23,1,24,7,25,1
A034258Write n! as a product of n numbers, n! = k(1)*k(2)*...*k(n) with k(1) <= k(2) <= ..., in all possible ways; a(n) = max value of k(1).
1 problem1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,8,8,8,9,9,10,10,10,10,11,11,12,12,12,12,12,12,13,13,13,14,14,15,
A034259Smallest m such that A034258(m) >= n.
1 problem1,4,9,14,16,20,24,27,32,34,38,40,46,49,51,57,58,62,65,68,72,77,80,84,87,90,93,94,100,104,108,111,114,115,118,125,125,128
A034463Maximal number of residue classes mod n such that no subset adds to 0.
1 problem0,1,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,8,9,9,9,9,9,9,9,9,10,9,10,10,10,10,1
A036236Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.
1 problem1,0,3,4700063497,6,19147,10669,25,9,2228071,18,262279,3763,95,1010,481,20,45,35,2873,2951,3175999,42,555,50,95921,27,174
A038133From a subtractive Goldbach conjecture: odd primes that are not cluster primes.
1 problem97,127,149,191,211,223,227,229,251,257,263,269,293,307,331,337,347,349,367,373,379,383,397,409,419,431,457,479,487,499,5
A038372Largest subset of integers [ 1...n ] such that no member divides two others.
1 problem1,2,2,3,4,4,5,6,6,7,8,8,9,10,10,11,12,12,13,14,14,15,16,16,17,18,19,20,21,21,22,22,22,23,24,24,25,26,26,27,28,28,29,30
A039651Number of iterations of f(x) = phi(x)+1 on n required to reach a prime.
1 problem1,0,0,1,0,1,0,1,1,1,0,1,0,1,2,2,0,1,0,2,1,1,0,2,2,1,1,1,0,2,0,1,2,1,3,1,0,1,3,1,0,1,0,2,3,1,0,1,1,2,3,3,0,1,1,3,1,1,0,1,
A045535Least negative pseudosquare modulo the first n odd primes.
1 problem7,23,71,311,479,1559,5711,10559,18191,31391,118271,366791,366791,2155919,2155919,2155919,6077111,6077111,98538359,120293
A045945Hexagonal matchstick numbers: a(n) = 3*n*(3*n+1).
1 problem0,12,42,90,156,240,342,462,600,756,930,1122,1332,1560,1806,2070,2352,2652,2970,3306,3660,4032,4422,4830,5256,5700,6162,6
A046693Size of smallest subset S of N={0,1,2,...,n} such that S-S=N, where S-S={abs(i-j) | i,j in S}.
1 problem1,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,11,11,11,11,11,11,11,12,12,12,12,12,12,
A046801Number of divisors of 2^n-1.
1 problem1,2,2,4,2,6,2,8,4,8,4,24,2,8,8,16,2,32,2,48,12,16,4,96,8,8,8,64,8,96,2,32,16,8,16,512,4,8,16,192,4,144,8,128,64,16,8,768
A048669The Jacobsthal function g(n): maximal gap in a list of all the integers relatively prime to n.
1 problem1,2,2,2,2,4,2,2,2,4,2,4,2,4,3,2,2,4,2,4,3,4,2,4,2,4,2,4,2,6,2,2,3,4,3,4,2,4,3,4,2,6,2,4,3,4,2,4,2,4,3,4,2,4,3,4,3,4,2,6,
A048892Start of n consecutive integers with distinct number of divisors.
1 problem1,1,4,9,45,76,270,2204,3718,95499,590890,16023339,16475964,1745175039,31287652672,347321438520,2620400333120,23991979183
A049108a(n) is the number of iterations of Euler phi function needed to reach 1 starting at n (n is counted).
1 problem1,2,3,3,4,3,4,4,4,4,5,4,5,4,5,5,6,4,5,5,5,5,6,5,6,5,5,5,6,5,6,6,6,6,6,5,6,5,6,6,7,5,6,6,6,6,7,6,6,6,7,6,7,5,7,6,6,6,7,6,
A050259Numbers k such that 2^k == 3 (mod k).
1 problem1,4700063497,3468371109448915,8365386194032363,10991007971508067
A051487Numbers k such that phi(k) = phi(k - phi(k)).
1 problem2,6,12,24,48,96,150,192,300,384,600,726,750,768,1200,1452,1500,1536,2310,2400,2904,3000,3072,3174,3750,4620,4800,5046,58
A051488Numbers k such that phi(k) < phi(k - phi(k)).
1 problem30,60,66,120,132,138,174,210,240,246,264,276,318,330,348,420,480,492,498,510,528,534,552,630,636,660,678,690,696,786,840
A051572a(1) = 5, a(n) = sigma(a(n-1)).
1 problem5,6,12,28,56,120,360,1170,3276,10192,24738,61440,196584,491520,1572840,5433480,20180160,94859856,355532800,1040179456,21
A051761Numbers that are simultaneously a sum of factorials of distinct integers and of the form a^b with b >= 2.
1 problem0,1,4,8,9,25,27,32,121,128,144,729,841,5041,5184,45369,46225,363609,403225,3674889,1401602635449
A053597Let prime(i) = i-th prime (A000040), let d(i) = prime(i+1)-prime(i) (A001223); a(n) = number of distinct numbers among d(n), d(n+1), d(n+2), ... before first duplicate is encountered.
1 problem2,1,2,2,2,2,3,3,2,3,3,2,3,2,1,2,3,3,3,3,2,3,4,3,2,2,2,3,2,5,4,3,2,3,2,1,2,2,1,3,2,3,2,3,2,1,3,2,3,4,3,3,2,1,1,2,3,5,4,4,
A053760Smallest positive quadratic nonresidue modulo p, where p is the n-th prime.
1 problem2,2,2,3,2,2,3,2,5,2,3,2,3,2,5,2,2,2,2,7,5,3,2,3,5,2,3,2,2,3,3,2,3,2,2,3,2,2,5,2,2,2,7,5,2,3,2,3,2,2,3,7,7,2,3,5,2,3,2,3,
A054377Primary pseudoperfect numbers: numbers k > 1 such that 1/k + sum 1/p = 1, where the sum is over the primes p | k.
1 problem2,6,42,1806,47058,2214502422,52495396602,5998279018951962402
A055265a(n) is the smallest positive integer not already in the sequence such that a(n)+a(n-1) is prime, starting with a(1)=1.
1 problem1,2,3,4,7,6,5,8,9,10,13,16,15,14,17,12,11,18,19,22,21,20,23,24,29,30,31,28,25,34,27,26,33,38,35,32,39,40,43,36,37,42,41,
A056604a(0)=1; thereafter a(n) = lcm(1, 2, 3, 4, ..., prime(n)).
1 problem1,2,6,60,420,27720,360360,12252240,232792560,5354228880,2329089562800,72201776446800,5342931457063200,219060189739591200
A057561Size of the largest set encompassing no {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, ...} (A003586).
1 problem1,2,2,3,4,5,5,6,7,7,8,8,9,10,11,11,12,13,13,14,14,15,16,17,17,18,18,19,20,21,21,22,22,23,24,25,25,26,26,27,28,29,30,30,3
A059233Number of rows in which n appears in Pascal's triangle A007318.
1 problem1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,
A060407Maximal number of pairwise edge-disjoint monochromatic K_3's in a K_n for any 2-coloring of the edges of K_n.
1 problem0,0,0,1,2,2,3,4,6
A060427Smallest prime p such that there are n strings of consecutive integers all having products = 1 mod p.
1 problem2,11,17,23,71,71,599,599,3011,27901,52163,778699,2374649,10428007
A060859Powerful numbers of the form k^2 - 1.
1 problem8,288,675,9800,235224,332928,1825200,11309768,384199200,592192224,4931691075,13051463048,221322261600,443365544448,86536
A061070Number of distinct values in the list of values of the Euler totient function {phi(j) : j=1..n}.
1 problem1,1,2,2,3,3,4,4,4,4,5,5,6,6,7,7,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,14,14,15,15,15,15,16,16,17,17,17,17,18,1
A061092a(0) = 1; for n>0, a(n) = smallest prime of the form k*a(n-1) + 1.
1 problem1,2,3,7,29,59,709,2837,22697,590123,1180247,9441977,169955587,2719289393,5438578787,32631472723,391577672677,15663106907
A062241Smallest integer >= 2 that is not the sum of 2 positive integers whose prime factors are all <= prime(n), the n-th prime.
1 problem3,7,23,71,311,479,1559,5711,10559,18191,31391,118271,366791,366791,2155919,2155919,2155919,6077111,6077111,98538359,1202
A062249a(n) = n + d(n), where d(n) = number of divisors of n, cf. A000005.
1 problem2,4,5,7,7,10,9,12,12,14,13,18,15,18,19,21,19,24,21,26,25,26,25,32,28,30,31,34,31,38,33,38,37,38,39,45,39,42,43,48,43,50,
A063980Pillai primes: primes p such that there exists an integer m such that m! + 1 == 0 (mod p) and p != 1 (mod m).
1 problem23,29,59,61,67,71,79,83,109,137,139,149,193,227,233,239,251,257,269,271,277,293,307,311,317,359,379,383,389,397,401,419,
A064113Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.
1 problem2,15,36,39,46,54,55,73,102,107,110,118,129,160,164,184,187,194,199,218,239,271,272,291,339,358,387,419,426,464,465,508,5
A064152Erdős primes: primes p such that all p-k! for 1 <= k! < p are composite.
1 problem2,101,211,367,409,419,461,557,673,709,769,937,967,1009,1201,1259,1709,1831,1889,2141,2221,2309,2351,2411,2437,2539,2647,
A064164EHS numbers: k such that there is a prime p satisfying k! + 1 == 0 (mod p) and p !== 1 (mod k).
1 problem8,9,13,14,15,16,17,18,19,20,21,22,23,24,26,29,30,31,32,33,34,35,36,38,39,40,42,43,44,45,47,48,49,50,51,52,53,54,55,56,57
A064491a(1) = 1, a(n+1) = a(n) + tau(a(n)), where tau(n) (A000005) is the number of divisors of n.
1 problem1,2,4,7,9,12,18,24,32,38,42,50,56,64,71,73,75,81,86,90,102,110,118,122,126,138,146,150,162,172,178,182,190,198,210,226,2
A066063Size of the smallest subset S of T={0,1,2,...,n} such that each element of T is the sum of two elements of S.
1 problem1,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12
A066766Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.
1 problem2,7,4,4,0,3,3,8,8,8,7,5,9,4,8,8,3,6,0,4,8,0,2,1,4,8,9,1,4,9,2,2,7,2,1,6,4,3,1,1,4,2,8,9,8,1,3,1,9,6,3,9,3,1,7,8,4,8,5,2,
A068063Maximum cardinality of a nondividing subset of {1, 2, ..., n}.
1 problem0,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
A068509a(n) = maximum length of a subset in {1,..,n} whose integers have pairwise LCM not exceeding n.
1 problem1,2,2,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10
A070003Numbers divisible by the square of their largest prime factor.
1 problem4,8,9,16,18,25,27,32,36,49,50,54,64,72,75,81,98,100,108,121,125,128,144,147,150,162,169,196,200,216,225,242,243,245,250,
A070089P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.
1 problem1,2,4,6,8,9,10,12,16,18,20,21,22,24,25,27,28,30,32,33,36,40,42,45,46,48,50,52,54,56,57,58,60,64,66,68,70,72,75,77,78,81,
A071626Number of distinct exponents in the prime factorization of n!.
1 problem0,1,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,
A071799Number of lattice paths in the lattice [0..2n] X [0..2n] which do not pass through the point (n,n).
1 problem2,34,524,7970,121252,1850380,28337976,435443490,6711230900,103711749284,1606464657096,24935144010764,387746052588104,603
A071870Numbers k such that gpf(k) > gpf(k+1) > gpf(k+2) where gpf(k) denotes the largest prime factor of k.
1 problem13,14,34,37,38,43,61,62,73,79,86,94,103,118,122,123,142,151,152,157,158,163,173,185,193,194,202,206,214,218,223,229,241,
A072207a(0) = 1; for n>0, a(n) = number of distinct sums of subsets of {1, 1/2, 1/3, 1/4, ..., 1/n} (allowing the empty subset).
1 problem1,2,4,8,16,32,52,104,208,416,832,1664,1856,3712,7424,9664,19328,38656,59264,118528,126976,224128,448256,896512,936832,18
A072937Least k such that prime(n) appears in factorization of k! + 1.
1 problem2,4,3,5,12,16,9,14,18,30,36,40,21,23,52,15,8,18,7,72,23,13,88,96,100,6,106,86,112,63,65,16,16,50,150,156,81,166,172,89,1
A073016Decimal expansion of Sum_{n>=1} 1/binomial(2n,n).
1 problem7,3,6,3,9,9,8,5,8,7,1,8,7,1,5,0,7,7,9,0,9,7,9,5,1,6,8,3,6,4,9,2,3,4,9,6,0,6,3,1,2,5,8,3,2,9,0,9,4,9,7,9,0,5,6,8,2,1,9,6,
A073101Number of integer solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.
1 problem0,0,1,1,2,5,5,6,4,9,7,15,4,14,33,22,4,21,9,30,25,22,19,45,10,17,25,36,7,72,17,62,27,22,59,69,9,29,67,84,7,77,12,56,87,39
A073944a(n) is the smallest m such that n-th prime divides m! + 1.
1 problem1,2,4,3,5,12,16,9,14,18,30,36,40,21,23,52,15,8,18,7,72,23,13,88,96,100,6,106,86,112,63,65,16,16,50,150,156,81,166,172,89
A074738Decimal expansion of 1-(1+log(log(2)))/log(2).
1 problem0,8,6,0,7,1,3,3,2,0,5,5,9,3,4,2,0,6,8,8,7,5,7,3,0,9,8,7,7,6,9,2,2,6,7,7,7,6,0,5,9,1,1,0,9,5,3,0,3,3,3,1,7,3,4,9,2,0,2,0,
A074741Sum of squares of gaps between consecutive primes.
1 problem1,5,9,25,29,45,49,65,101,105,141,157,161,177,213,249,253,289,305,309,345,361,397,461,477,481,497,501,517,713,729,765,769
A074781Primes of the form p*2^k + 1 for any k and any prime p.
1 problem3,5,7,11,13,17,23,29,41,47,53,59,83,89,97,107,113,137,149,167,173,179,193,227,233,257,263,269,293,317,347,353,359,383,38
A075245x-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075246 and A075247.
1 problem1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,14,13,13,14,14,
A075246y-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The x and z components are in A075245 and A075247.
1 problem4,3,4,7,15,7,10,16,34,13,18,29,61,21,30,46,96,31,43,67,139,43,60,92,190,57,78,121,249,73,100,154,316,91,124,191,391,111,
A075247Largest possible z-value of an integer solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z. The x and y components are in A075245 and A075246.
1 problem12,6,20,42,210,42,90,240,1122,156,468,812,3660,420,510,2070,9120,930,1806,4422,19182,1806,2100,8372,35910,3192,9048,1452
A075248Number of solutions (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.
1 problem0,1,2,1,1,3,5,9,6,3,12,5,18,15,10,5,21,11,22,18,15,8,55,30,15,20,43,20,45,5,24,35,23,36,53,10,21,52,62,6,62,12,73,69,16,
A076259Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).
1 problem1,1,2,1,1,3,1,2,1,1,2,2,2,1,1,3,3,1,1,2,1,1,2,1,1,2,1,1,3,1,4,2,2,2,1,1,2,1,3,1,1,2,1,1,2,1,3,1,1,3,1,2,1,1,2,2,2,1,1,2,
A076336(Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite.
1 problem78557,271129,271577,322523,327739,482719,575041,603713,903983,934909,965431,1259779,1290677,1518781,1624097,1639459,1777
A076445The smaller of a pair of powerful numbers (A001694) that differ by 2.
1 problem25,70225,130576327,189750625,512706121225,13837575261123,99612037019889,1385331749802025,3743165875258953025,10114032809
A076871Sum of two powerful numbers (definition (1), A001694).
1 problem2,5,8,9,10,12,13,16,17,18,20,24,25,26,28,29,31,32,33,34,35,36,37,40,41,43,44,45,48,50,52,53,54,57,58,59,61,63,64,65,68,7
A076872a(n) = number of numbers <= n that are the sum of two squarefull numbers.
1 problem0,1,1,1,2,2,2,3,4,5,5,6,7,7,7,8,9,10,10,11,11,11,11,12,13,14,14,15,16,16,17,18,19,20,21,22,23,23,23,24,25,25,26,27,28,28
A078515Numbers n such that A053597(n) sets a new record.
1 problem1,7,23,30,94,219,279,773,1856,3724,6999,7000,19205,184163,280103,849876,1870722,3570761,4114341,11271072,55282774,682560
A080200Numbers that do not occur as differences between terms of the Mian-Chowla sequence A005282.
1 problem33,88,98,99,105,106,112,126,130,132,134,150,152,154,156,162,163,165,170,176,184,188,198,205,214,215,217,220,222,228,234,
A083550Product of 2 consecutive prime differences of two successive terms of A001223.
1 problem2,4,8,8,8,8,8,24,12,12,24,8,8,24,36,12,12,24,8,12,24,24,48,32,8,8,8,8,56,56,24,12,20,20,12,36,24,24,36,12,20,20,8,8,24,1
A087280Solutions n of max(m+d(m))=n+2 for m<n; d(m) is the number of divisors of m.
1 problem5,8,10,12,24
A090162Values of binomial(Fibonacci(2k)*Fibonacci(2k+1),Fibonacci(2k-1)*Fibonacci(2k)-1).
1 problem1,3003,61218182743304701891431482520
A091963a(n) is the smallest gcd of two interior numbers on row n of Pascal's triangle ("interior" means that the 1's at the ends of the rows are excluded).
1 problem2,3,2,5,2,7,2,3,2,11,3,13,2,3,2,17,2,19,4,3,2,23,3,5,2,3,4,29,6,31,2,3,2,5,4,37,2,3,5,41,6,43,4,3,2,47,3,7,2,3,4,53,2,5,
A092487a(n) = least k such that {n+1, n+2, n+3, ... n+k} has a subset the product of whose members with n is a square.
1 problem0,4,5,0,5,6,7,7,0,8,11,8,13,7,9,0,17,9,19,10,7,11,23,8,0,13,8,12,29,12,31,13,11,17,13,0,37,19,13,10,41,14,43,11,15,23,47
A092670a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0<x_1<...<x_k<=n.
1 problem1,1,1,1,1,2,2,2,2,2,2,3,3,3,6,6,6,11,11,22,22,22,22,41,41,41,41,114,114,200,200,200,363,363,566,852,852,852,852,1655,165
A092671Numbers n such that there exists a solution to the equation 1 = 1/x_1 + ... + 1/x_k (for any k), 0 < x_1 < ... < x_k = n.
1 problem1,6,12,15,18,20,24,28,30,33,35,36,40,42,45,48,52,54,55,56,60,63,65,66,70,72,75,76,77,78,80,84,85,88,90,91,95,96,99,100,1
A094708Size of the smallest set hitting all {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27,...} (A003586).
1 problem0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,13,14,14,15,15,15,16,16,
A097847Triangle read by rows: T(n,k) = minimal number of terms needed to write k/n (for 1 <= k <= n) as a sum of unit fractions.
1 problem1,1,1,1,2,1,1,1,2,1,1,2,2,3,1,1,1,1,2,2,1,1,2,3,2,3,3,1,1,1,2,1,2,2,3,1,1,2,1,2,2,2,3,3,1,1,1,2,2,1,2,2,3,3,1,1,2,2,2,3,
A097849Maximal entry in row n of A097847.
1 problem1,1,2,2,3,2,3,3,3,3,4,3,4,4,3,4,5,3,4,3,4,4,5,3,4,4,4,4,5,4,5,4,4,5,4,4,5,5,5,4,5,4,5,4,4,5,5,4,5,5,5,5,5,4,5,4,5,5,5,4,
A098565Numbers that appear as binomial coefficients exactly 6 times.
1 problem120,210,1540,7140,11628,24310,61218182743304701891431482520
A103309Smallest prime primitive root of n that is less than n, or 0 if none exists.
1 problem0,0,0,2,3,2,5,3,0,2,3,2,0,2,3,0,0,3,5,2,0,0,7,5,0,2,7,2,0,2,0,3,0,0,3,0,0,2,3,0,0,7,0,3,0,0,5,5,0,3,3,0,0,2,5,0,0,0,3,2,
A105206Number of edges in a pancyclic graph on n+2 vertices with the fewest possible edges.
1 problem3,5,6,8,9,10,12,13,14,15,16,17,19,20,21,22,23,24,25,26
A109925Number of primes of the form n - 2^k.
1 problem0,0,1,2,1,2,2,1,2,1,2,1,2,1,3,0,1,2,3,1,4,0,2,1,2,0,3,0,1,1,2,1,3,1,3,0,2,1,4,0,1,1,2,1,5,0,2,1,3,0,3,0,1,1,3,0,2,0,1,1,
A110177Number of solutions 0<k<n to the equation sigma(n) = sigma(k) + sigma(n-k), where sigma is the sum of divisors function.
1 problem0,0,2,0,0,0,0,2,2,2,0,0,0,0,2,0,0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,4,2,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,2,0,0,0,4,2,4,0,0,0,
A110566a(n) = lcm{1,2,...,n}/denominator of harmonic number H(n).
1 problem1,1,1,1,1,3,3,3,1,1,1,1,1,1,1,1,1,3,3,15,45,45,45,15,3,3,1,1,1,1,1,1,11,11,11,11,11,11,11,11,11,77,77,7,7,7,7,7,1,1,1,1,
A113827Initial terms associated with the arithmetic progressions of primes in A005115.
1 problem2,2,3,5,5,7,7,199,199,199,110437,110437,4943,31385539,115453391,53297929,3430751869,4808316343,8297644387,214861583621,5
A115645Powerful(1) numbers (A001694) that are sums of distinct factorials.
1 problem1,8,9,25,27,32,121,128,144,729,841,864,5041,5184,40328,41067,45369,45387,46208,46225,363609,403225,3674889,43954688,6230
A116446Let Sq(n) denote the square grid consisting of all lattice points (x,y) such that x,y are in {0,1,...,n}. a(n) is the minimum number t such that there are t of the (n+1)^2 lattice points in Sq(n) so t
1 problem1,4,4,4,6,6,7,8,8,8
A119425Primitive terms of the sequence A119357, i.e., of the sequence of those values of n for which the number of distinct nonzero sums of distinct divisors of n is less than 2^tau(n) - 1.
1 problem6,20,28,45,63,70,88,99,104,105,110,117,130,154,165,170,182,195,231,238,255,266,272,273,285,286,304,322,345,357,368,374,3
A120414a(0)=0, a(1)=1; thereafter a(n) = ceiling((3/2)^(n-3)*n*(n-1)).
1 problem0,1,2,6,18,45,102,213,426,821,1538,2820,5075,8996,15743,27247,46709,79405,133996,224640,374400,620715,1024180,1682580,27
A121048a(n) = n + phi(n), where phi is the Euler totient function.
1 problem2,3,5,6,9,8,13,12,15,14,21,16,25,20,23,24,33,24,37,28,33,32,45,32,45,38,45,40,57,38,61,48,53,50,59,48,73,56,63,56,81,54,
A121359Greatest prime factor of pyramidal number A000292(n).
1 problem2,5,5,7,7,7,5,11,11,13,13,13,7,17,17,19,19,19,11,23,23,23,13,13,13,29,29,31,31,31,17,17,17,37,37,37,19,41,41,43,43,43,23
A123556Number of elements in longest possible arithmetic progression of primes with difference n.
1 problem2,3,2,3,2,5,1,3,2,3,2,5,1,3,2,2,2,4,1,3,2,2,1,4,1,2,2,3,2,6,1,2,1,3,2,4,1,3,2,3,2,5,1,2,2,2,1,5,1,3,2,2,1,4,1,2,2,2,2,6,
A129515Numbers m such that binomial(2*m, m) has the same prime factors as binomial(2*k, k) for some k > m.
1 problem87,199,237,467,607,967,1127,1319,1483,1903,1943,2012,2047,2287,2348,2359,2464,2479,2495,2507,2623,2645,2719,3349,3467,35
A131628Maximal size of an n-distance set in the plane.
1 problem1,3,5,7,9,12,13
A135311A greedy sequence of prime offsets.
1 problem0,2,6,8,12,18,20,26,30,32,36,42,48,50,56,62,68,72,78,86,90,96,98,102,110,116,120,128,132,138,140,146,152,156,158,162,168
A137245Decimal expansion of Sum_{p prime} 1/(p * log p).
1 problem1,6,3,6,6,1,6,3,2,3,3,5,1,2,6,0,8,6,8,5,6,9,6,5,8,0,0,3,9,2,1,8,6,3,6,7,1,1,8,1,5,9,7,0,7,6,1,3,1,2,9,3,0,5,8,6,0,0,3,0,
A140462Turan's upper bound on the number of triangles of a simplicial complex of dimension two for which every minimal non-face has three vertices.
1 problem0,0,0,1,3,7,14,23,36,54,75,102,136,174,220,275,335,405,486,573,672,784,903,1036,1184,1340,1512,1701,1899,2115,2350,2595,
A141399Positive integers k such that the distinct primes that divide k or k+1 form a set of consecutive primes. In other words, k is included if and only if k*(k+1) is contained in sequence A073491.
1 problem1,2,3,5,8,9,14,15,20,24,35,80,125,224,384,440,539,714,1715,2079,2400,3024,4374,9800,12375,123200,194480,633555
A143823Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.
1 problem1,2,4,7,13,22,36,57,91,140,216,317,463,668,962,1359,1919,2666,3694,5035,6845,9188,12366,16417,21787,28708,37722,49083,63
A147807Partial sums of A147810(n) = tau(n^2 + 1)/2.
1 problem1,2,4,5,7,8,11,13,15,16,18,20,24,25,27,28,32,35,37,38,42,44,48,49,51,52,56,58,60,62,66,69,73,75,77,78,82,85,87,88,91,93,
A154554Primes p such that m=p-1 is the least number such that p divides m!+1.
1 problem2,3,5,13,17,31,37,41,53,73,89,97,101,107,113,151,157,167,173,181,197,211,223,229,241,281,283,313,331,337,349,353,373,409
A155085a(n) = n + sum of divisors of n.
1 problem2,5,7,11,11,18,15,23,22,28,23,40,27,38,39,47,35,57,39,62,53,58,47,84,56,68,67,84,59,102,63,95,81,88,83,127,75,98,95,130,
A156816Decimal expansion of the positive root of the equation 13x^4 - 7x^2 - 581 = 0.
1 problem2,6,3,8,1,5,8,5,3,0,3,4,1,7,4,0,8,6,8,4,3,0,3,0,7,5,6,6,7,4,4,4,1,3,0,4,8,8,8,0,5,0,2,2,0,1,0,3,1,8,3,5,9,7,3,7,0,7,8,7,
A156989Largest size of a subset of {1,2,3}^n that does not contain any combinatorial lines (i.e., strings formed by 1, 2, 3, and at least one instance of a wildcard x, with x then substituted for 1, 2, or 3,
1 problem1,2,6,18,52,150,450
A160559Minimal covering numbers.
1 problem12,80,90,210,280,378,448,1386,1650,2200,2464,5346,9750,11264,11466,13000,14994,18954,20384,23166,26656,27846,30294,31122
A171081Van der Waerden numbers w(3, n).
1 problem9,18,22,32,46,58,77,97,114,135,160,186,218,238,279,312,349
A175155Numbers m satisfying m^2 + 1 = x^2 * y^3 for positive integers x and y.
1 problem0,682,1268860318,1459639851109444,2360712083917682,86149711981264908618,4392100110703410665318,8171493471761113423918890
A175769Maximum cardinality of isosceles sets in E^n.
1 problem3,6,8,11,17,28,30,45
A180058Smallest number occurring in exactly n rows of Pascal's triangle.
1 problem2,6,120,3003
A181740Number of sequences of length n over {1, -1} with Erdős discrepancy <= 2.
1 problem1,2,4,6,12,18,28,44,88,100,152,240,370,556,882,750,1500,2250,2784,4284,6438,6062,9526,14856,22944,26164,39528,35122,5480
A182237Numbers occurring exactly in 2 rows of Pascal's triangle.
1 problem6,10,15,20,21,28,35,36,45,55,56,66,70,78,84,91,105,126,136,153,165,171,190,220,231,252,253,276,286,300,325,330,351,364,3
A185661Smallest set containing 1 and closed under the operations x->2x+1, x->3x+1, x->6x+1.
1 problem1,3,4,7,9,10,13,15,19,21,22,25,27,28,31,39,40,43,45,46,51,55,57,58,61,63,64,67,76,79,81,82,85,87,91,93,94,103,111,115,11
A186736Maximum sum of relatively prime integers no larger than n.
1 problem1,3,6,8,13,13,20,24,30,30,41,41,54,54,55,63,80,80,99,99,103,103,126,126,146,146,159,164,193,193,224,235,235,235,238,238,
A192881Number of terms for the shortest Egyptian fraction representation of 1 starting with 1/n.
1 problem1,3,5,8,10,11,13,15,17,19,21,23,25,26,28,30
A193429a(n) = minimum value of the largest element of a nonempty set of positive integers > n such that their product is equal to n!, or 0 if no such set exists.
1 problem1,0,0,6,24,12,10,20,16,28,25,22,33,30,28,28,39,35,36,44,44,42,44,50,50,50,57,57,56,58,65,64,64,72,72,70,75,80,80,78,80,8
A194259Number of distinct prime factors of p(1)*p(2)*...*p(n), where p(n) is the n-th partition number.
1 problem0,1,2,3,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,8,9,9,10,11,12,13,14,15,16,17,17,18,19,20,21,21,21,22,23,24,25,27,28,30,31,32,
A194260A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)*p(2)*...*p(n) and p(n) is the n-th partition number.
1 problem-1,-1,-1,-1,-1,-1,-2,-3,-4,-5,-6,-7,-7,-8,-9,-10,-11,-12,-13,-13,-14,-14,-14,-15,-15,-15,-15,-15,-15,-15,-15,-15,-16,-16
A210184Number of distinct residues of all factorials mod prime(n).
1 problem2,3,4,5,6,10,12,12,17,19,21,26,29,26,31,35,37,41,42,39,44,49,55,59,59,65,71,75,63,73,80,82,90,90,104,86,103,104,107,111,
A213253a(n) = smallest k such that highest prime factor of m(m+1)...(m+k-1) is > n if m > n.
1 problem1,2,3,3,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,
A214583Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.
1 problem3,4,6,8,12,14,18,20,24,30,32,38,42,48,54,60,62,68,72,80,84,90,98,108,110,132,138,140,150,180,182,198,252,318,360,398,468
A217693Numbers of distinct integers obtained from summing up subsets of {1, 1/2, 1/3, ..., 1/n}.
1 problem1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
A219429Highest prime primitive root (less than p) for the n-th prime p. (or 0 if none exists).
1 problem0,2,3,5,7,11,11,13,19,19,17,19,29,29,43,41,47,59,61,67,59,59,79,83,83,89,101,103,103,107,109,127,131,109,139,109,151,149
A226521Triangle read by rows: T(n,k) = smallest prime == k (mod n) if gcd(k,n)=1, otherwise 0, for n >= 2, 1 <= k < n.
1 problem3,7,2,5,0,3,11,2,3,19,7,0,0,0,5,29,2,3,11,5,13,17,0,3,0,5,0,7,19,2,0,13,5,0,7,17,11,0,3,0,0,0,7,0,19,23,2,3,37,5,17,7,19
A227988Decimal expansion of Sum_{n >= 1} sigma_1(n)/n!.
1 problem3,5,2,7,0,0,0,4,7,1,8,5,2,9,5,2,8,2,9,7,6,1,5,3,6,7,9,1,7,6,9,3,2,6,2,0,3,7,6,3,5,6,4,3,4,4,9,5,2,4,0,8,2,7,7,6,0,5,7,1,
A227989Decimal expansion of Sum_{n >= 1} sigma_2(n)/n!.
1 problem6,3,4,0,0,9,6,6,6,8,8,9,2,1,7,1,6,3,8,8,2,9,9,6,5,9,9,4,0,0,7,5,0,4,6,0,7,8,6,3,6,4,4,3,3,5,5,9,8,9,0,1,7,8,5,4,3,9,9,6,
A229487Conjectured greatest number that converges to prime(n) under the iteration x -> phi(x) + 1, where phi is Euler's totient function.
1 problem2,6,12,30,22,138,60,54,46,58,62,174,498,510,94,106,118,4314,134,142,1038,158,166,276,420,250,206,214,750,1758,254,262,27
A231255a(n) is the smallest integer t such that every length-t walk from the origin (0,0) taking steps of either (0,1) or (1,0) is guaranteed to have n points that are collinear.
1 problem0,1,4,9,29,97
A234813Number of distinct integers of the form i+(i+1)+(i+2)+...+j, for 1 <= i <= j <= n.
1 problem1,3,5,9,12,16,21,27,33,40,47,55,63,70,77,89,101,110,123,134,146,159,171,186,200,214,229,245,260,275,293,312,329,349,369,
A237695Maximum length of a +- 1 sequence of discrepancy n.
1 problem0,11,1160
A245762Maximal number of edges in a C_4 free subgraph of the n-cube.
1 problem1,3,9,24,56,132
A256435First differences of sums of two squares.
1 problem1,1,2,1,3,1,1,3,3,1,1,2,5,1,3,3,2,2,1,3,1,4,4,1,2,1,5,3,3,1,3,4,1,1,6,1,1,3,4,1,7,1,2,1,3,2,3,4,3,1,4,1,3,3,2,6,1,7,1,1,
A256519Composites c for which an integer 1 < k < c exists such that (c-k)! == -1 (mod c).
1 problem25,121,169,437,551,667,721,1037,1159,1273,1349,1403,1541,1769,1943,2209,2329,2363,2419,3071,3713,4087,5041,5111,7313,835
A256936Decimal expansion of Sum_{k>=1} phi(k)/2^k, where phi is Euler's totient function.
1 problem1,3,6,7,6,3,0,8,0,1,9,8,5,0,2,2,3,5,0,7,9,0,5,0,8,1,4,6,2,1,3,0,8,8,1,3,9,0,7,4,8,9,1,9,9,8,9,6,2,7,9,4,8,5,2,9,5,6,5,9,
A259180Amicable pairs.
1 problem220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,14595,17296,18416,63020,76084,66928,66992,67095,71145,
A262153Decimal expansion of Sum_{p prime} 1/(2^p-1), a prime equivalent of the Erdős-Borwein constant.
1 problem5,1,6,9,4,2,8,1,9,8,0,5,6,4,0,3,8,4,2,4,0,5,1,6,6,0,8,4,7,9,8,5,6,2,7,7,9,7,8,5,4,6,9,4,7,9,1,3,0,9,1,2,4,1,6,5,0,2,8,0,
A263958Prime Riesel numbers p that are not Mersenne primes such that 2*p is a noncototient.
1 problem509203,2554843,9203917,9545351,10645867,11942443,14608183,15627133,15811777,16413457,21013423,21465637,25792993,30622663
A263996Smallest possible cardinality of the union of the set of pairwise sums and the set of pairwise products from a set of n positive integers.
1 problem1,4,7,11,15,20,26,30,36,44,49,57,64,71,80,86,96,104,112,121,131,141,150,160,169,179,190,200,212,222,235,248,260,272,283,
A276661Least k such that there is a set S in {1, 2, ..., k} with n elements and the property that each of its subsets has a distinct sum.
1 problem0,1,2,4,7,13,24,44,84,161,309
A280992Squarefree triangular numbers that are products of consecutive primes.
1 problem1,3,6,15,105,210,255255
A284783Numbers k such that k and k + 5040 have the same number of divisors.
1 problem11,19,22,37,38,39,41,46,47,51,55,57,58,59,61,62,65,67,68,73,74,76,78,79,87,88,91,92,99,102,104,107,113,114,115,116,118,1
A287116Nonsquare integers that cannot be represented in the form 4M-d, where (a*b)|M and d|(a+b) for some positive integers a,b.
1 problem288,336,4545
A289280a(n) is the least integer k > n such that any prime factor of k is also a prime factor of n.
1 problem4,9,8,25,8,49,16,27,16,121,16,169,16,25,32,289,24,361,25,27,32,529,27,125,32,81,32,841,32,961,64,81,64,49,48,1369,64,81,
A322144a(n) = Sum_{i=1..phi(n)-1} (r(i+1)-r(i))^2 where r(1) = 1 < ... < n-1 = r(phi(n)) are the phi(n) integers relatively prime to n.
1 problem0,0,1,4,3,16,5,12,11,24,9,36,11,32,29,28,15,56,17,52,39,48,21,76,31,56,41,68,27,128,29,60,59,72,57,116,35,80,69,108,39,1
A322292a(n) = Max_{c composite, c < n} (c + least prime factor of c).
1 problem6,6,8,8,10,12,12,12,14,14,16,18,18,18,20,20,22,24,24,24,26,30,30,30,30,30,32,32,34,36,36,40,40,40,40,42,42,42,44,44,46,4
A325864Number of subsets of {1..n} of which every subset has a different sum.
1 problem1,2,4,7,13,22,36,56,91,135,211,307,446,625,882,1194,1677,2238,3031,4001,5460,6995,9302,11921,15424,19554,25032,31005,391
A327153Number of divisors d of n such that sigma(d)*d is equal to n.
1 problem1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
A327909a(n) is the smallest start of a run of n or more integers having a prime factor greater than n.
1 problem2,5,13,19,55,65,113,151,151,226,364,406,736,736,1057,1057,1409,1409,2059,2059,2313,2313,2313,2313,2313,2313,2313,6007,69
A330244Weird numbers m (A006037) such that sigma(m)/m > sigma(k)/k for all weird numbers k < m, where sigma(m) is the sum of divisors of m (A000203).
1 problem70,10430,1554070,5681270,6365870
A331372Decimal expansion of Sum_{k>=1} 1/(2^k - 3).
1 problem3,4,3,6,7,3,4,3,3,1,8,1,7,6,9,0,1,8,5,4,4,4,8,2,8,3,3,3,8,1,2,4,1,2,0,6,1,8,8,8,0,7,1,7,6,4,8,6,7,8,3,8,4,8,6,5,1,1,0,5,
A331373Decimal expansion of Sum_{k>=2} 1/(k! - 1).
1 problem1,2,5,3,4,9,8,7,5,5,6,9,9,9,5,3,4,7,1,6,4,3,3,6,0,9,3,7,9,0,5,7,9,8,9,4,0,3,6,9,2,3,2,2,0,8,3,3,2,0,1,3,4,1,7,0,6,3,8,3,
A332077Square array of sunflower numbers Sun(m,n) = minimal number of distinct sets of cardinality <= m such that there is a sunflower with at least n sets among them, read by falling antidiagonals; m, n >=
1 problem1,2,1,3,2,1,4,7,2,1,5,11,21,2,1,6,21
A333230Positions of weak ascents in the sequence of differences between primes.
1 problem1,2,3,5,7,8,10,13,14,15,17,20,22,23,26,28,29,31,33,35,36,38,39,41,43,45,46,49,50,52,54,55,57,60,61,64,65,67,69,70,71,73,
A333231Positions of weak descents in the sequence of differences between primes.
1 problem2,4,6,9,11,12,15,16,18,19,21,24,25,27,30,32,34,36,37,39,40,42,44,46,47,48,51,53,54,55,56,58,59,62,63,66,68,72,73,74,77,8
A335277First index of strictly increasing prime quartets.
1 problem7,13,22,28,49,60,64,69,70,75,78,85,89,95,104,116,122,123,144,148,152,155,173,178,182,195,201,206,212,215,219,225,226,230
A339378Let n be a positive integer. For each prime divisor p of n, consider the highest power of p which does not exceed n. The sum a(n) of these powers is defined as the power-sum of n.
1 problem0,2,3,4,5,7,7,8,9,13,11,17,13,15,14,16,17,25,19,21,16,27,23,25,25,29,27,23,29,68,31,32,38,49,32,59,37,51,40,57,41,66,43,
A339465Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.
1 problem19,31,37,43,61,67,73,79,103,109,127,139,157,163,181,199,223,229,241,271,277,283,307,313,337,349,367,373,379,397,409,433,
A343101Pairs of integers (k, m) ordered by m with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1.
1 problem2,8,6,48,14,224,30,960,75,1215,62,3968,126,16128,254,65024,510,261120,1022,1046528,2046,4190208,4094,16769024,8190,67092
A343507a(n) is the smallest nonnegative integer k such that (2*k)! / (k+n)!^2 is an integer.
1 problem0,208,3475,8174,252965,3648835,72286092,159329607,2935782889
A344005a(n) = smallest positive m such that n divides the oblong number m*(m+1).
1 problem1,1,2,3,4,2,6,7,8,4,10,3,12,6,5,15,16,8,18,4,6,10,22,8,24,12,26,7,28,5,30,31,11,16,14,8,36,18,12,15,40,6,42,11,9,22,46,1
A352287Numbers k such that, for every prime p dividing k, k has a nontrivial divisor which is congruent to 1 (mod p).
1 problem1,12,24,30,36,48,56,60,72,80,90,96,105,108,112,120,132,144,150,160,168,180,192,210,216,224,240,252,264,270,280,288,300,3
A359747Numbers k such that k*(k+1) has in its canonical prime factorization mutually distinct exponents.
1 problem1,3,4,7,8,16,24,27,31,48,63,71,72,107,108,124,127,199,242,243,256,400,431,432,499,512,576,647,783,863,967,971,1024,1151,
A360659a(n) is the minimum sum of a completely multiplicative sign sequence of length n.
1 problem0,1,0,-1,0,-1,0,-1,-2,-1,0,-1,-2,-3,-4,-3,-2,-3,-4,-5,-4,-5,-4,-5,-6,-5,-6,-7,-8,-9,-8,-9,-8,-7,-8,-7,-6,-7,-8,-7,-8,-9,
A362137Smallest size of an n-paradoxical tournament built as a directed Paley graph.
1 problem1,3,7,19,67,331,1163
A363069Size of the largest subset of {1,2,...,n} such that no two elements sum to a perfect square.
1 problem1,1,1,2,2,3,4,4,4,4,5,5,6,6,6,7,8,8,8,8,9,9,10,10,11,11,12,12,12,13,13,13,13,14,14,14,15,15,16,16,17,17,18,18,18,19,19,1
A364132a(n) is the smallest positive integer such that from the set {1, 2, ..., a(n)} one can choose an increasing sequence (s(1), s(2), ..., s(n)) in which every segment has a unique sum of elements.
1 problem1,2,4,5,7,10,12,13,15,18,21,24,25,29,30,33,36,38,41,47,50,52
A364153a(n) is the smallest positive integer such that from the set {1, 2, ..., a(n)} one can choose a sequence (s(1), s(2), ..., s(n)) in which every segment has a unique sum.
1 problem1,2,3,5,6,7,9,10,12,13,14,17,18
A365339Length of the longest subsequence of 1,...,n on which the Euler totient function phi A000010 is nondecreasing.
1 problem1,2,3,4,5,5,6,6,7,7,8,8,9,9,10,11,12,12,13,13,13,13,14,14,14,14,15,15,16,16,17,17,17,17,18,18,19,19,19,19,20,20,21,21,21
A365474a(n) = A365339(10^n).
1 problem1,7,34,193,1276,9656,78562,664643,5761519,50847598
A367090Numbers that cannot be written as a sum of distinct powers of 3 and distinct powers of 4.
1 problem62,63,143,144,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,23
A371134Decimal expansion of Sum_{squarefree k>=1} k / 2^k.
1 problem1,6,9,7,9,0,7,8,1,9,7,7,9,6,2,5,0,6,4,4,6,4,2,4,0,8,9,9,6,5,3,4,7,8,9,1,8,4,3,6,3,5,1,5,3,1,8,8,6,2,4,7,2,6,3,4,0,6,9,9,
A372040Smallest k such that there is an n-element subset of {1, 2, ..., k} that does not contain a (nonempty) subset that sums to a square.
1 problem2,3,5,8,12,18,22,34,40,62,76,85,134
A372306Cardinality of the largest subset of {1,...,n} such that no three distinct elements of this subset multiply to a square.
1 problem1,2,3,4,5,5,6,6,6,7,8,8,9,10,10,10,11,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,2
A373114Cardinality of the largest subset of {1,...,n} such that no odd number of terms from this subset multiply to a square.
1 problem0,1,2,2,3,3,4,5,5,5,6,7,8,9,9,9,10,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,29,3
A373178Cardinality of the largest subset of {1,...,n} such that no five distinct elements of this subset multiply to a square.
1 problem1,2,3,4,5,5,6,7,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,2
A373319Denominator of the asymptotic density of numbers that are unitarily divided by n.
1 problem1,4,9,8,25,18,49,16,27,25,121,36,169,98,225,32,289,54,361,50,147,242,529,72,125,169,81,196,841,225,961,64,1089,289,1225,
A375071Smallest k such that Product_{i=1..k} (n+i) divides Product_{i=1..k} (n+k+i), or 0 if there is no such k.
1 problem1,5,4,207,206,2475,984,8171,8170,45144,45143,3648830,3648829,7979077,7979076,58068862,58068861,255278295,255278294,10195
A375077Smallest k such that Product_{i=0..n} (k-i) divides C(2*k,k).
1 problem2,2480,8178,45153,3648841,7979090,101130029,339949252,1019547844,17609764994,1070858041585,5048891644646,18253129921842,
A375081Smallest k>n such that the denominator of Sum {i=n..k} (1/i) is larger than the denominator of Sum {i=n..k+1} (1/i).
1 problem5,5,5,17,17,14,14,14,14,14,32,34,34,34,27,27,27,27,23,23,27,51,51,51,51,44,44,44,44,44,39,39,39,39,39,44,74,74,74,74,74,
A377587a(n) is the smallest odd integer m with m-2^k not squarefree for all 1<=k<=n.
1 problem11,29,533,849,434977,10329791,28819433,129747557,6915752957,2569472629649,23373845739407,60690478781437
A380791For a positive rational x, let k(x) be the smallest positive integer such that all k >= k(x) have a partition into distinct parts with reciprocal sum equal to x. The n-th term in this sequence is equa
1 problem2,2,2,1,2,4,5,5,7,7,5,12,18,22,32,38,41,48,57,76,82,74,97,117,155,170,194,228,277,306,332,430,473,483,510
A381776Empty polygon numbers: a(n) is the smallest number of points in the plane (with no three of them collinear) such that an empty convex n-gon cannot be avoided.
1 problem3,5,10,30
A382395Number of maximum sized subsets of {1..n} such that every pair of distinct elements has a different difference.
1 problem1,1,1,3,2,6,14,2,10,26,60,110,4,22,68,156,320,584,8,24,80,206,504,1004,1910,3380,10,34,98,282,760,1618,3334,6360,11482,2
A382397Minimum size of a maximal subset of {1..n} such that every pair of distinct elements has a different difference.
1 problem0,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
A383044Numbers m such that phi(m) + phi(m+phi(m)) = m where phi is the Euler totient function.
1 problem4,6,8,10,12,14,16,20,24,28,32,40,48,56,64,70,80,94,96,112,128,140,160,188,192,224,256,280,320,376,384,448,512,560,640,75
A385316Smallest number that is the sum of 3 cubes of primes in exactly n different ways.
1 problem24,185527,8627527,999979163,10588881419
A385657Number of nonisomorphic maximally dense unit-distance graphs on n vertices.
1 problem1,1,1,1,1,4,1,3,1,1,2,1,1,2,1,1,7,16,3,1,5
A386439Decimal expansion of the maximal density of a set of positive integers free of subsets of the form {n, 2n, 3n}.
1 problem8,0,0,9,6,5,7,5,5,0,0,6,5,5,8,9,8,9,0,9,0,4,2,0,3,2,6,3,8,8,0,8,2,4,1
A386620a(n) is the smallest integer k > 2*n such that Product_{i=1..n} (k - i) has no prime factor p in n < p < 2*n.
1 problem3,6,9,20,13,21,21,22,65,220,51,338,133,321,339,340,113,114,368,550,805,2691,1884,2664,7653,7654,36887,36888,21234,21235,
A386893Minimal number of Farey fractions in between two fractions that are not similarly ordered.
1 problem2,3,3,4,3,3,4,5,4,5,5,5,5,6,6,7,6,7,7,8,7,7,8,8,8,9,9,10,9,10,10,11,10,11,11,12,11,12,12,14,12,13,13,15,13,13,14,15,14,1
A386978Numbers k such that the k-th prime gap contains an integer whose least prime factor is greater than or equal to the length of the prime gap.
1 problem2,3,5,7,10,13,15,17,20,21,26,28,32,33,34,35,37,39,41,42,43,45,47,49,52,53,54,55,57,60,61,64,66,68,69,72,73,74,77,79,81,8
A387053a(n) is the least k such that n is the sum of a prime and k powers of 2.
1 problem0,0,1,0,1,0,1,1,1,0,1,0,1,1,2,0,1,0,1,1,2,0,1,1,2,1,2,0,1,0,1,1,1,1,2,0,1,1,2,0,1,0,1,1,2,0,1,1,2,1,2,0,1,1,2,1,2,0,1,0,
A387054Elements of A388654 that do not lie in A070003.
1 problem24,48,120,168,360,528,840,960,1155,1368,1680,1683,1848,1850,2208,2210,2600,2736,2737,2808,3024,3250,3480,3720,4224,4488,
A387184Numbers d such that a!*b!*c!*d! is a perfect square for some 1<=a<b<c<d.
1 problem6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,48,49,50,51,52,54,55,56,57,60,62,63,
A387502Number of debut sums of initial subsequences of the divisors > 1 of n.
1 problem0,1,1,1,1,1,1,1,1,1,0,3,1,1,1,1,0,2,1,2,2,1,0,1,0,0,1,1,1,3,0,1,1,1,0,4,1,0,0,2,0,3,1,1,1,1,0,3,1,2,0,2,0,2,0,1,2,0,0,3,
A387503Total number of distinct sums of initial subsequences of the divisors > 1 of positive integers up to n.
1 problem0,1,2,3,4,5,6,7,8,9,9,12,13,14,15,16,16,18,19,21,23,24,24,25,25,25,26,27,28,31,31,32,33,34,34,38,39,39,39,41,41,44,45,46
A387543a(n) is the size of the largest subset of {1, 2, ..., n} containing n in which any two numbers share a prime factor.
1 problem1,1,1,2,1,3,1,4,3,5,1,6,1,7,5,8,1,9,1,10,7,11,1,12,5,13,9,14,1,15,1,16,11,17,7,18,1,19,13,20,1,21,1,22,15,23,1,24,7,25,1
A387584Size of largest subset of {1, ..., n} such that any 4 elements (a <= b <= c <= d) whose product is square implies a*d = b*c.
1 problem1,2,3,3,4,5,6,6,6,7,8,8,9,10,10,10,11,11,12,12,13,14,15,15,15,16,16,16,17,17,18,18,19,20,21,21,22,23,24,24,25,25,26,26,2
A387591Primes prime(k) such that (k+1)*prime(k) < k*prime(k+1).
1 problem3,5,7,13,19,23,31,37,43,47,53,61,67,73,79,83,89,97,103,109,113,131,139,151,157,167,173,181,199,211,233,241,251,257,263,2
A387635a(n) = Sum_{k=0..n-1} binomial(2*n, k)^2.
1 problem0,1,17,262,3985,60626,925190,14168988,217721745,3355615450,51855874642,803232328548,12467572005382,193873026294052,30196
A387688Number of solutions to n = 2^r + 3^s + 2^t * 3^u where r, s, t and u are nonnegative integers.
1 problem0,0,1,2,3,5,4,5,4,4,7,6,8,7,6,4,6,5,8,7,8,4,8,0,6,5,6,4,10,3,7,5,6,7,9,6,12,6,6,4,10,3,9,7,6,4,8,0,8,2,6,4,8,0,5,2,5,3,8
A387698Irregular triangle read by rows in which row n is the maximum clique containing n in the gcd-graph of {1, 2, ..., n}. If more than one such clique exists, choose the lexicographically earliest.
1 problem1,2,3,2,4,5,2,4,6,7,2,4,6,8,3,6,9,2,4,6,8,10,11,2,4,6,8,10,12,13,2,4,6,8,10,12,14,3,6,9,12,15,2,4,6,8,10,12,14,16,17,2,4
A387704Size of the maximal subset S of {1,2,...,n} such that for all a, b, c in S not necessarily distinct, a+b+c is unique up to permutation.
1 problem0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
A387858Sum of squares of the multiplicities of pairwise distances among the vertices of a regular n-gon.
1 problem9,20,50,81,147,208,324,425,605,756,1014,1225,1575,1856,2312,2673,3249,3700,4410,4961,5819,6480,7500,8281,9477,10388,1177
A387864Numbers r for which there are at least two integers strictly between prime(r) and prime(r+1), all of whose prime factors are less than prime(r+1) - prime(r).
1 problem4,9,11,15,24,30,34,37,46,47,53,62,66,92,99,114,137,146,150,154,168,172,180,189,205,217,242,259,263,274,278,283,293,295,3
A387897Decimal expansion of the connective constant of the square lattice.
1 problem2,6,3,8,1,5,8,5,3,0,3,2,7
A388302a(n) = is the smallest m >= 1 such that n! = b_1*...*b_t with b_1 < ... < b_t and m = b_t - b_1.
1 problem1,1,2,2,2,2,4,6,7,6,9,9,9,12,14,12,15,16,17,18,19,19,20,21,22,24,24,24,27,27,29,30,30,32,34,33,34,34,37,37,38,38,39,41,4
A388654Elements of an interval of natural numbers whose product is divisible by the square of the largest prime factor.
1 problem4,8,9,16,18,24,25,27,32,36,48,49,50,54,64,72,75,81,98,100,108,120,121,125,128,144,147,150,162,168,169,196,200,216,225,24
A388850Initial term of first maximal bad interval of width n, i.e., initial term of the first run of exactly n+1 consecutive integers in A388654; or 0 if no such interval exists.
1 problem4,8,48,1680,76725,332925,7474752,4541154,75047565
A388851Numbers c such that a! * b! * c! is a perfect square for some 1 <= a < b < c.
1 problem4,6,8,9,10,16,18,20,24,25,28,30,32,35,36,45,49,50,54,63,64,70,72,77,80,81,96,98,100,112,120,121,125,126,128,140,144,150,
A389100Elements of a non-singleton interval of natural numbers whose product is divisible by the square of the largest prime factor of the product.
1 problem8,9,24,25,48,49,50,120,121,168,169,242,243,288,289,360,361,528,529,675,676,840,841,960,961,1155,1156,1368,1369,1444,1445
A389117Decimal expansion of the sum of the distinct entries of 1/A055204(n)^(1/2).
1 problem3,7,0,9,7,5,1,2,3,3,8,9,9,2,2,2,3,6,8,8,8,7,9,6,1,4,6,6,8,4,2,0,7,8,8,8,2,1,7,4,4,3,6,4,5,0,6,4,1,6,9,7,6,6,8,0,9,7,8,9,
A389148Composite numbers e>6 for which there is no 1<=a<b<c<d<e for which a!b!c!d!e! is a perfect square.
1 problem527,611,713,731,779,893,923,1003,1037,1271,1273,1343,1349,1357,1411,1469,1591,1643,1679,1781,1919,1927,1943,1957,2033,20
A389182Maximum cardinality of a subset of {1,...,n} in which all sums a+b of elements a<=b are distinct except possibly one.
1 problem1,2,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12,
A389241Maximum number of distinct consecutive sums of a permutation of [n].
1 problem0,1,3,6,9,13,19,25,32,39,47,56,66,77,89,100
A389244Powerful part of n*(n+1): a(n) = A057521(n*(n+1)).
1 problem1,1,4,4,1,1,8,72,9,1,4,4,1,1,16,16,9,9,4,4,1,1,8,200,25,27,108,4,1,1,32,32,1,1,36,36,1,1,8,8,1,1,4,36,9,1,16,784,1225,25
A389313a(n) is the smallest m such that for every red-blue edge-coloring of the graph K_{m} there exists either a red or a blue n-cycle; Ramsey number r(C_n, C_n).
1 problem6,6,9,8,13,11,17,14,21,17,25,20,29,23,33,26,37,29,41,32,45,35,49,38,53,41,57,44,61,47,65,50,69,53,73,56,77,59,81,62,85,6
A389335a(n) is the smallest m such that for every red-green-blue edge-coloring of the graph K_{m} there exists at least one red, green or blue n-cycle; Ramsey number r(C_n, C_n, C_n).
1 problem17,11,17,12,25,16
A389360Smallest m >= 2*n such that binomial(m,n) is a multiple of m-i for all 0<=i<n, but one.
1 problem4,6,9,12,75,30,70,56,2403,280,3465,210,793,4732,3213,1456,31110,612,67203,145540,464646,2640,476938,21000,86550,234026,1
A389396Numbers k such that binomial(2*k,k) is divisible by (k+1)^2*(k+2)^2.
1 problem208,458,987,1220,1455,1597,1889,2012,2144,2330,2477,2663,2991,3353,3415,3430,3439,3475,3476,3551,3563,3568,3617,3625,372
A389479a(n) is the Frobenius number for the set { binomial(m,k), k=1..m-1 } where m = A024619(n).
1 problem49,1043,989,20669,12907,99007,67031,700319,7054529,750397,124807499,7125065,578549,1935378079,37337700047,41645613,18836
A389544a(n) is the smallest integer greater than a(n-1) such that all consecutive products in a(1)..a(n) are distinct, with a(1) = 2.
1 problem2,3,4,5,7,8,9,10,11,13,14,15,17,18,19,21,22,23,25,26,27,28,29,30,31,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,5
A389646Maximum number of edges that need to be removed from a triangle-free graph on n vertices to make it bipartite.
1 problem0,0,0,0,1,1,1,2,2,4,4,5,6,7,9,9,10,12,13,16,16,17,20
A389676a(n) = Min_{0<i<n} (prime(n-i) + prime(n+i)).
1 problem7,10,16,20,26,32,40,48,54,62,70,78,88,96,104,114,120,130,140,150,156,166,174,182,192,202,212,220,236,244,252,264,276,286
A389677a(n) = A389676(n) - 2*prime(n).
1 problem1,0,2,-2,0,-2,2,2,-4,0,-4,-4,2,2,-2,-4,-2,-4,-2,4,-2,0,-4,-12,-10,-4,-2,2,10,-10,-10,-10,-2,-12,-8,-8,-14,-4,-8,-8,0,-10
A389680Numbers k for which no i > 1 exists such that k + i is composite and its least prime factor exceeds i^2.
1 problem1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,34,35,36,37,38,39,40,41,42,43,44,45,
A389713a(1) = 3, a(2) = 5, a(n) is the smallest prime such that a(n) - a(n-1) + 1 is in the sequence.
1 problem3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,101,103,107,109,113,131,137,139,149,151,157,163,167,17
A389784a(n) is the maximum size of a subset A of {1,...,n} such that no element in A is the average of a subset of A with cardinality at least 2.
1 problem1,2,2,3,4,4,4,4,4,5,6,6,6,6,6,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,1
A389839Smallest even number which cannot be written as the difference of two consecutive numbers which are relatively prime to the primorial prime(n)#.
1 problem6,8,12,16,20,28,32,42,48,60,68
A389975Maximum cardinality of a set of disjoint congruence classes with distinct moduli, each at most n.
1 problem1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,10,10,10,10,10,11,11,12,12,12,12,
A390181Numbers not able to be represented in the form c*p^2+b for p prime, c >= 1 and 0 <= b < p.
1 problem1,2,3,6,7,14,15,22,23,30,31,34,35,39,42,43,58,59,62,66,67,70,71,86,87,94,95,106,107,111,114,115,134,138,139,142,143,158,
A390187Minimum number of distinct consecutive sums of a permutation of [n].
1 problem0,1,3,5,7,10,13,17,20,25,30,33,39,44,50,56,63,70,77
A390246a(n) is the least integer k such that there exist n distinct integers b_1, ..., b_n with n < b_i < n+k and b_i is divisible by i for 1 <= i <= n.
1 problem2,3,4,6,6,9,9,11,13,15,15,17,16,19,20,25,24,27,26,29,30,31,30,33,36,38,40,43,42,46,45,50,49,48,50,55,54,55,58,60,59,61,6
A390256Minimum size of maximum clique or independent set of a graph on n vertices.
1 problem0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4
A390257Minimum size of maximum regular induced subgraph of a graph on n vertices.
1 problem0,1,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5
A390380Integers k which cannot be written in the form x^2 + y^2 - z^2, where x, y, z are integers and x^2, y^2, z^2 <= k.
1 problem3,6,11,15,22,27,35,38,42,55,59,66,78,83,87,95,110,118,123,131,143,150,187,210,222,227,255,262,266,278,299,303,323,326,39
A390393a(n) is the maximum size of a subset A of {1,...,n} such that Sum_{k in S} 1/k != 1 for all subsets S of A.
1 problem0,1,2,3,4,4,5,6,7,8,9,10,11,12,12,13,14,15,16,16,17,18,19,19,20,21,22,22,23,24,25,26,26,27,28,29,30,31,32,33,34,34,35,36
A390394a(n) is the maximum size of a subset A of {1,...,n} such that there are no solutions to 1/a = 1/b_1 + ... + 1/b_k with distinct a, b_1, ..., b_k in A.
1 problem1,2,3,4,5,5,6,7,8,9,10,10,11,12,12,13,14,15,16,16,17,18,19,19,20,21,22,22,23,23,24,25,26,27,27,27,28,29
A390395a(n) is the maximum size of a subset S of {1,...,n} such that there are no solutions to 1/a = 1/b + 1/c for distinct a,b,c in S.
1 problem1,2,3,4,5,5,6,7,8,9,10,10,11,12,13,14,15,16,17,18,19,20,21,21,22,23,24,25,26,26,27,28,29,30,31,32,33,34,35,35,36,36,37,3
A390642a(n) is the smallest integer k such that the number of sums a(i) + a(j) <= k for i <= j < n is less than k - n + 1.
1 problem1,3,5,9,13,17,24,31,38,45,53,61,75,87,97,112,124,139,147,175,182,205,219,242,265,277,309,313,349,378,386,430,445,478,480
A390645Triangle read by rows: T(n,r) is maximal such that there exists a family F of subsets of {1,...,n} of size T(n,r) such that the intersection of no two sets in F has r elements.
1 problem1,1,2,2,3,4,4,4,7,8,8,6,11,15,16,16,11,16,26,31,32,32,23,22,42,57,63,64,64,45,37,64,99,120,127,128,128,94,67,93,163,219,
A390769Least even positive integer k that does not appear in the first n prime gaps.
1 problem2,4,4,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,16,16,16
A390813a(n) is the size of the largest Sidon subset of the first n positive perfect squares.
1 problem1,2,3,4,5,6,6,7,8,9,9,9,10,10,11,12,12,13,13,13,14,14,14,15,16,17,17,17,17,18,19,19,19,20,20,20,21,21,22,22,22,22,23,24,
A390848Complement of A389544.
1 problem1,6,12,16,20,24,35,56,60,72,90,110,120,140,143,182,210,255,280,306,342,352,399,420,462,504,506,575,650,702,715,720,756,8
A390919Number of graphs with n vertices that have no induced regular subgraph of order 5 or greater.
1 problem1,2,4,11,31,130,728,6027,66308,818276,8336902,45933753,79888458,23814804,512906,954,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
A391118Let B_n = {b_1 < b_2 < ...} be the set of those integers in [n, n^2] which have a divisor in (n, 2n). a(n) = max(b_(i+1) - b_i).
1 problem3,3,3,3,4,4,4,4,5,6,6,5,6,6,6,6,6,6,8,8,8,8,8,7,8,8,8,8,8,8,8,8,8,8,8,8,8,7,9,9,10,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,1
A391431Numerator of k + a/k where n = k^2 + 2*a + 1 and -k <= a <= k.
1 problem1,1,3,2,2,7,5,8,3,3,13,10,7,11,15,4,4,21,17,22,9,23,19,24,5,5,31,26,16,27,11,28,17,29,35,6,6,43,37,44,19,45,13,46,20,47,
A391490Denominator of k + a/k where n = k^2 + 2a + 1 and -k <= a <= k.
1 problem1,1,2,1,1,3,2,3,1,1,4,3,2,3,4,1,1,5,4,5,2,5,4,5,1,1,6,5,3,5,2,5,3,5,6,1,1,7,6,7,3,7,2,7,3,7,6,7,1,1,8,7,4,7,8,7,2,7,8,7,
A391592Maximum size of a subset S of {1..n} such that all subset sums of {1/k : k in S} are distinct.
1 problem1,2,3,4,5,5,6,7,8,9,10,10,11,12,12,13,14,14,15,15,15,16,17,17,18,19,20,20,21,21,22,23,23,24,24,25,26,27,28,28,29,29,30,3
A391599Minimum size of an intersecting family of n-sets such that every set of size at most n-1 is disjoint from at least one member of the family.
1 problem1,3,6,9,13
A391750Maximum length of an increasing sequence, bounded by n, in which the largest prime divisors of the elements form a decreasing sequence.
1 problem1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,
A392019If binomial(n,k) = u*v where the only primes dividing u are <= k and the only primes dividing v are > k, then a(n) is the least k such that u > n^2, or 0 if such a k doesn't exist.
1 problem0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,11,0,13,0,0,8,13,0,17,12,13,13,5,17,7,19,9,12,11,11,7,11,17,15,14,17,17,19,13,13,19,23,17,
A392165Indices of record values in A392164.
1 problem1,5,19,23,37,41,59,87,101,105,113,131,151,159,167,195,203,239,259,303,307,403,451,499,517,553,573,609,645,701,719,787,80
A392342Numbers that are not the sum of at most four cubefull numbers.
1 problem5,6,7,12,13,14,15,20,21,22,23,31,38,39,46,47,53,58,69,77,79,85,95,101,103,111,175,196,212,228,231,247,327,444,458,490,60
A392343Numbers that are not the sum of at most five 4-full numbers.
1 problem6,7,8,9,10,11,12,13,14,15,21,22,23,24,25,26,27,28,29,30,31,37,38,39,40,41,42,43,44,45,46,47,52,53,54,55,56,57,58,59,60,6
A392636Number of graphs with n vertices that have no induced regular subgraph of order 6 or greater.
1 problem1,2,4,11,34,148,960,10390,188560,5317230,202396620,8905369148,384098286140,13129756210164
A393168Integers k which can be written in the form x^2 + y^2 - z^2, where x, y, z are integers and x^2, y^2, z^2 <= k.
1 problem0,1,2,4,5,7,8,9,10,12,13,14,16,17,18,19,20,21,23,24,25,26,28,29,30,31,32,33,34,36,37,39,40,41,43,44,45,46,47,48,49,50,51
A393584a(n) is the minimum, over all partitions {X, Y} of {1..2n} with |X| = |Y|, of the maximum number of pairs (x, y) with the same difference x - y for x in X and y in Y.
1 problem1,1,2,2,3,3,3,4,4,5,5,5,6,6,6,7,7,8,8,8,9,9,10,10,10,11,11,11,12,12,13,13,13
A394400Number of graphs with n vertices that have no induced regular subgraph of order 4 or greater.
1 problem1,2,4,7,11,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
A394462Number of graphs with n vertices that have no induced regular subgraph of order 6.
1 problem1,2,4,11,34,148,964,10472,191776,5524670,219302174,10333796899,493296884096,19658348081642
A394539Number of graphs with n vertices that have no induced regular subgraph of order 5.
1 problem1,2,4,11,31,136,792,7185,94893,1714430,37216434,854671213,18369802688,328662169364,4236467418682,29440587191035,80145694
A394563Least integer a(n) such that every graph on a(n) vertices has an induced regular subgraph of order at least n.
1 problem1,2,5,7,17
A394564Least integer a(n) such that every graph on a(n) vertices has an induced regular subgraph of order n.
1 problem1,2,6,8,21
A394573Number of graphs with n vertices that have no induced regular subgraph of order 4.
1 problem1,2,4,7,12,12,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
A394574a(n) is the greatest k such that every graph on n vertices has an induced regular subgraph of order k.
1 problem0,1,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5
A394930Number of graphs with n vertices that have no induced regular subgraph of order 7.
1 problem1,2,4,11,34,156,1038,12246,269646,11453460,907948002,127924347122,30302185606487
A394933Number of graphs with n vertices that have no induced regular subgraph of order 7 or greater.
1 problem1,2,4,11,34,156,1038,12226,268920,11361262,885194426,119298229792,25716285392622