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#30Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]number theory · 3 Vela attempts (new OEIS term) · AI tried (unverified) · A143824,A227590,A003022#124For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that\[\sum_{…number theory · 10 Vela attempts (verified reduction) · AI tried (unverified) · N/A#242For every $n>2$ there exist distinct integers $1\leq x<y<z$ such that\[\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]Erdős-Straus conjecture · number theory · 1 Vela attempt (verified reduction) · AI tried (unverified) · A073101,A075245,A075246,A075247,A075248,A287116#295Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Is it true that\[\lim_{N\to \infty} k(N)-(e-1)N=…number theory · 1 Vela attempt (verified reduction) · AI tried (unverified) · A192881#306Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n…number theory · 26 Vela attempts (improved bound) · AI tried (unverified) · N/A#307Are there two finite sets of primes $P,Q$ such that\[1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?\]number theory · 3 Vela attempts (verified reduction) · AI tried (unverified) · N/A#319What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that\[\sum_{n\in A}\frac{\delta_n}{n}=0\]and\[\sum_{n\in A'}\frac{\delta_n}{n}\neq…number theory · 4 Vela attempts (partial proof) · AI tried (unverified) · possible#366Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$.number theory · 2 Vela attempts (verified reduction) · AI tried (unverified) · A060355#376Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?number theory · 1 Vela attempt (obstruction map) · AI tried (unverified) · A030979#396Is it true that for every $k$ there exists $n$ such that\[\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?\]number theory · 1 Vela attempt (verified reduction) · AI tried (unverified) · A375077#417Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq x\}\]and\[V(x)=\#\{\phi(m) \leq x : 1\leq m\}.\]Does $\lim V(x)/V'(x)$ exist? Is it $>1$?number theory · 2 Vela attempts (partial proof) · AI tried (unverified) · A264810,A061070#458Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,\[[1,\ldots,p_{k+1}-1]< p_k[1,\ldots,p_k]?\]number theory · 6 Vela attempts (verified reduction) · AI tried (unverified) · A056604#463Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that\[n+f(n)<m<n+p(m)?\](Here $p(m)$ is the least prime factor…number theory · 3 Vela attempts (partial proof) · AI tried (unverified) · possible#470Call $n$ weird if $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.Are there any odd weird numbers? Are there infinitely many primitive weird numb…number theory · 1 Vela attempt (obstruction map) · AI tried (unverified) · A006037,A002975#488Let $A$ be a finite set and\[B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.\]Is it true that, for every $m>n\geq \max(A)$,\[\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\…number theory · 12 Vela attempts (verified reduction) · AI tried (unverified) · N/A#647Let $\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that\[\max_{m<n}(m+\tau(m))\leq n+2?\]number theory · 2 Vela attempts (obstruction map) · AI tried (unverified) · A062249,A087280#686Can every integer $N\geq 2$ be written as\[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\]for some $k\geq 2$ and $m\geq n+k$?number theory · 1 Vela attempt (partial proof) · AI tried (unverified) · N/A#699Is it true that for every $1\leq i<j\leq n/2$ there exists some prime $p\geq i$ such that\[p\mid \textrm{gcd}\left(\binom{n}{i}, \binom{n}{j}\right)?\]number theory · 11 Vela attempts (verified reduction) · AI tried (unverified) · N/A#931Let $k_1\geq k_2\geq 3$. Are there only finitely many $n_2\geq n_1+k_1$ such that\[\prod_{1\leq i\leq k_1}(n_1+i)\textrm{ and }\prod_{1\leq j\leq k_2}(n_2+j)\]have the same prime factors?number theory · 5 Vela attempts (verified reduction) · AI tried (unverified) · N/A#971Let $p(a,d)$ be the least prime congruent to $a\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$,\[p(a,d) > (1+c)\phi(d)\log d\]for $\gg \phi(d)$ many values of $a$?number theory · 2 Vela attempts (partial proof) · AI tried (unverified) · A226521#1056Let $k\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\ldots,I_k$ such that\[\prod_{n\in I_i}n \equiv 1\pmod{p}\]for all $1\leq i\leq k$?number theory · 9 Vela attempts (verified reduction) · AI tried (unverified) · A060427#1093For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $…number theory · 7 Vela attempts (verified reduction) · AI tried (unverified) · N/A#1094For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions.number theory · 5 Vela attempts (verified reduction) · AI tried (unverified) · N/A#148Let $F(k)$ be the number of solutions to\[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\]where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.number theory · 1 Vela attempt (verified reduction) · AI tried (unverified) · A076393,A006585#301Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}= \frac{1}{b_1}+\cdots+\frac{1}{b_k}\]with distinct $a,b_1,\ldots,b_k\in A$?Estimate $…number theory · 5 Vela attempts (improved bound) · AI tried (unverified) · A390394#302Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}= \frac{1}{b}+\frac{1}{c}\]with distinct $a,b,c\in A$?Estimate $f(N)$. In particular, …number theory · 22 Vela attempts (improved bound) · AI tried (unverified) · A390395#684For $0\leq k\leq n$ write\[\binom{n}{k} = uv\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let $f(n)$ be the smallest $k$ such that $u>n^2$. G…number theory · 1 Vela attempt (partial proof) · AI tried (unverified) · A392019,possible#700Let\[f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).\] Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$. Are there infinitely ma…number theory · 2 Vela attempts (partial proof) · AI tried (unverified) · A091963,possible#1If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then\[N \gg 2^{n}.\]number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A276661#7Is there a distinct covering system all of whose moduli are odd?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#10Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2?number theory · 2 Vela attempts (partial proof) · AI tried (unverified) · A387053#14Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.Is it true that for all $\epsilon>0$ …number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A143824,possible#33Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible value of\[\limsup \frac{\lvert A\cap\{1,\ldots,N\}\rv…number theory · 1 Vela attempt (partial proof) · AI tried (unverified) · N/A#39Is there an infinite Sidon set $A\subset \mathbb{N}$ such that\[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\]for all $\epsilon>0$?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#41Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that\[\liminf \frac{\lvert A\cap \{1,\…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#44Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#254Let $A\subseteq \mathbb{N}$ be such that\[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty\]and\[\sum_{n\in A} \{ \theta n\}=\infty\]for every $\theta\in (0,1)$, …number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#273Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#312Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subse…number theory · 1 Vela attempt (obstruction map) · AI tried (unverified) · N/A#313Are there infinitely many solutions to\[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\]where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A054377#321What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A384927,A391592#324Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#326Does there exist $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ which is a minimal basis of order $2$ (i.e. every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has t…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#329Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can\[\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}\]be?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · possible#340Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no n…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A080200,A005282#342With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i<j\leq n$.What can be said about this sequence? Do infinitel…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A002858#346Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that $A\backslash B$ is complete for any finite subset $B$ and $A\backslash B$ is not complete for any infinite subset $B$. (Here 'complete'…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#348For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that $A$ remains complete after removing any $m$ elements, but $A$ is not complete after rem…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#357Let $1\leq a_1<\cdots <a_k\leq n$ be integers such that all sums of the shape $\sum_{u\leq i\leq v}a_i$ are distinct. Let $f(n)$ be the maximal such $k$.How does $f(n)$ grow? Is $f(n)=o(n)$?number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A364132,A364153,possible#359Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of th…segmented numbers · number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A002048#361Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an ir…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · possible#377Is there some absolute constant $C>0$ such that\[\sum_{p\leq n}1_{p\nmid \binom{2n}{n}}\frac{1}{p}\leq C\]for all $n$ (where the summation is restricted to primes $p\leq n$)?number theory · 1 Vela attempt (partial proof) · AI tried (unverified) · N/A#387Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$?number theory · 1 Vela attempt (partial proof) · AI tried (unverified) · N/A#409How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?number theory · 2 Vela attempts (partial proof) · AI tried (unverified) · A039651,A229487#477Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $n\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(k) : k\in\mat…number theory · 1 Vela attempt (obstruction map) · AI tried (unverified) · N/A#489Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let\[B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.\]If $B=\{b_1<b_2<\cdots\}$ then is it true that\[\…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#535Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $…number theory · 1 Vela attempt (partial proof) · AI tried (unverified) · possible#881Let $A\subset\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\subset A$ is any infinite set then $A\backslash B$ is not a basis of order $k$. Must there exist an…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#885For integer $n\geq 1$ we define the factor difference set of $n$ by\[D(n) = \{\lvert a-b\rvert : n=ab\}.\]Is it true that, for every $k\geq 1$, there exist integers $N_1<\cdots<N_k$ such that\…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#930Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then\[\prod_{1\leq i\leq r}\prod_{m\in I_i}m\]i…number theory · 2 Vela attempts (partial proof) · AI tried (unverified) · N/A#1062Let $f(n)$ be the size of the largest subset $A\subseteq \{1,\ldots,n\}$ such that there are no three distinct elements $a,b,c\in A$ such that $a\mid b$ and $a\mid c$. How large can $f(n)$ be? Is $\li…number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A038372#1113A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every $2^km+1$ is divisible by…Sierpinski numbers · number theory · 1 Vela attempt (honest null) · AI tried (unverified) · A076336#3If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions?number theory · AI tried (unverified) · A003002,A003003,A003004,A003005#9Let $A$ be the set of all odd integers $\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?number theory · AI tried (unverified) · A006286#11Is every large odd integer $n$ the sum of a squarefree number and a power of 2?number theory · AI tried (unverified) · A001220,A377587#15Is it true that\[\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\]converges, where $p_n$ is the sequence of primes?number theory · AI tried (unverified) · N/A#17Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$?cluster primes · number theory · AI tried (unverified) · A038133#18We call $m$ practical if every integer $1\leq n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice.Are there infinitely…practical numbers · number theory · AI tried (unverified) · A005153#25Let $1\leq n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n<…number theory · AI tried (unverified) · N/A#28If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$.number theory · AI tried (unverified) · N/A#32Is there a set $A\subset\mathbb{N}$ such that\[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\]and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$? Can the boun…number theory · AI tried (unverified) · N/A#36Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two equal parts, so that $\lvert A\rvert=\lvert B\…minimum overlap problem · number theory · AI tried (unverified) · A393584,possible#38Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b\in B$ such th…number theory · solved · AI tried (unverified) · N/A#40For what functions $g(N)\to \infty$ is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}\]implies $\limsup 1_A\ast 1_A(n)=\infty$?number theory · AI tried (unverified) · N/A#42Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\subset \{1,\ldots,N\}$ there is another Sidon set $B\subset \{1,\ldots,N\}$ of size $M$ such that $(…number theory · solved · AI tried (unverified) · N/A#43If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that\[ \binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1),\]where $f(N)$ is …number theory · solved · AI tried (unverified) · A143824,A227590,A003022,possible#50Schoenberg proved that for every $c\in [0,1]$ the density of\[\{ n\in \mathbb{N} : \phi(n)<cn\}\]exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ e…number theory · AI tried (unverified) · N/A#51Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to…number theory · AI tried (unverified) · A002202,A014197#52Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\]sum-product problem · number theory · AI tried (unverified) · A263996#66Is there $A\subseteq \mathbb{N}$ such that\[\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}\]exists and is $\neq 0$?number theory · AI tried (unverified) · N/A#68Is\[\sum_{n\geq 2}\frac{1}{n!-1}\]irrational?number theory · AI tried (unverified) · A331373#123Let $a,b,c\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\geq 0$), none of which divide any other?number theory · AI tried (unverified) · N/A#126Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$?number theory · AI tried (unverified) · possible#137We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$. Let $k\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?number theory · AI tried (unverified) · N/A#145Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$,\[\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha\]exists?number theory · AI tried (unverified) · A005117#208Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$,\[s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?\]Is it true that\[s_{n+1}-s_n \leq (1+o(1))\f…number theory · AI tried (unverified) · A005117,A076259#218Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.number theory · AI tried (unverified) · A333230,A333231,A064113#233Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that\[\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.\]number theory · AI tried (unverified) · A074741#234For every $c\geq 0$ the density $f(c)$ of integers for which\[\frac{p_{n+1}-p_n}{\log n}< c\]exists and is a continuous function of $c$.number theory · AI tried (unverified) · N/A#236Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?number theory · AI tried (unverified) · A039669,A109925#238Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the difference between any two is $>c_2$?number theory · AI tried (unverified) · N/A#243Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that\[\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1\]and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large $n\geq 1$,\[ …number theory · AI tried (unverified) · A000058#244Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?number theory · AI tried (unverified) · N/A#247Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that\[\limsup \frac{a_n}{n}=\infty.\]Is\[\sum_{n=1}^\infty \frac{1}{2^{a_n}}\]transcendental?number theory · AI tried (unverified) · N/A#249Is\[\sum_n \frac{\phi(n)}{2^n}\]irrational? Here $\phi$ is the Euler totient function.number theory · AI tried (unverified) · A256936#251Is\[\sum \frac{p_n}{2^n}\]irrational? (Here $p_n$ is the $n$th prime.)number theory · AI tried (unverified) · A098990#252Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is\[\sum \frac{\sigma_k(n)}{n!}\]irrational?number theory · AI tried (unverified) · A227988,A227989,possible#276Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence…number theory · AI tried (unverified) · N/A#279Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$?number theory · AI tried (unverified) · N/A#282Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced b…number theory · AI tried (unverified) · N/A#283Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for all $n\geq 1$. Is it true that, for all sufficientl…number theory · solved · AI tried (unverified) · A380791#288Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that\[\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?\]number theory · AI tried (unverified) · N/A#289Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or adjacent, with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq…number theory · AI tried (unverified) · N/A#304For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $N(b)=\ma…number theory · AI tried (unverified) · A097847,A097849#317Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with\[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < …number theory · AI tried (unverified) · N/A#318Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that\[\sum_{n\in S}\frac{f(n)}…number theory · solved · AI tried (unverified) · N/A#323Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that\[f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}\]for all $\e…number theory · AI tried (unverified) · possible#325Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that\[f_{k,3}(x) \gg x^{3/k}\]or even $\gg_\epsilon x^{3/k-\epsilo…number theory · AI tried (unverified) · possible#330Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?number theory · solved · AI tried (unverified) · N/A#332Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gap…number theory · AI tried (unverified) · N/A#341Let $A=\{a_1<\cdots<a_k\}$ be a finite set of positive integers and extend it to an infinite sequence $\overline{A}=\{a_1<a_2<\cdots \}$ by defining $a_{n+1}$ for $n\geq k$ to be the l…number theory · AI tried (unverified) · N/A#347Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with\[\lim \frac{a_{n+1}}{a_n}=2\]such that\[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\]has density $1$ for e…number theory · solved · AI tried (unverified) · N/A#349For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t\alpha…number theory · AI tried (unverified) · N/A#351Let $p(x)\in \mathbb{Q}[x]$ with positive leading coefficient. Is it true that\[A=\{ p(n)+1/n : n\in \mathbb{N}\}\]is strongly complete, in the sense that, for any finite set $B$,\[\left\{\sum_{n\in X…number theory · solved · AI tried (unverified) · N/A#354Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset\[\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\…number theory · AI tried (unverified) · N/A#358Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to\[n=\sum_{u\leq i\leq v}a_i.\]Is there such an $A$ for which $f(n)\to \infty$ as $n\to \infty…number theory · solved · AI tried (unverified) · possible#364Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?number theory · AI tried (unverified) · A060355,A076445#371Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)<P(n+1)$ has density $1/2$.number theory · AI tried (unverified) · A070089#373Show that the equation\[n! = a_1!a_2!\cdots a_k!,\]with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions.number theory · AI tried (unverified) · A003135#375Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$ are all composite then there are distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$ for $1\leq i\leq k$?number theory · AI tried (unverified) · N/A