Let $F_{k} (N)$ be the size of the largest $A \subseteq {1, \dots, N}$ such that the product of no $k$ many distinct elements of $A$ is a square. Is $F_{5} (N) = (1 - o (1)) N$ ? More generally, is $F_{2 k + 1} (N) = (1 - o (1)) N$ ?

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A013928 — Number of (positive) squarefree numbers < n.0,1,2,3,3,4,5,6,6,6,7,8,8,9,10,11,11,12,12,13,13,14,15,16,16,16,17,17,17,18,19,20,20,21,22,23,23,24,25,26,26,27,28,29,29 A028391 — a(n) = n - floor(sqrt(n)).0,0,1,2,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,19,20,20,21,22,23,24,25,26,27,28,29,30,30,31,32,33,34,35,36,37,3 A143301 — Decimal expansion of the Hall-Montgomery constant.1,7,1,5,0,0,4,9,3,1,4,1,5,3,6,0,6,5,8,6,0,4,3,9,9,7,1,5,5,5,2,1,2,1,0,9,6,2,2,2,6,2,9,0,4,2,2,9,5,5,0,8,4,1,7,1,4,2,1,1,A360659 — a(n) is the minimum sum of a completely multiplicative sign sequence of length n.0,1,0,-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,A372306 — Cardinality of the largest subset of {1,...,n} such that no three distinct elements of this subset multiply to a square.1,2,3,4,5,5,6,6,6,7,8,8,9,10,10,10,11,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,2 A373114 — Cardinality of the largest subset of {1,...,n} such that no odd number of terms from this subset multiply to a square.0,1,2,2,3,3,4,5,5,5,6,7,8,9,9,9,10,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,29,3 A373178 — Cardinality of the largest subset of {1,...,n} such that no five distinct elements of this subset multiply to a square.1,2,3,4,5,5,6,7,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,2 A373319 — Denominator of the asymptotic density of numbers that are unitarily divided by n.1,4,9,8,25,18,49,16,27,25,121,36,169,98,225,32,289,54,361,50,147,242,529,72,125,169,81,196,841,225,961,64,1089,289,1225,

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#786Let

ϵ > 0

. Is there some set

A \subset N

of density

> 1 - ϵ

such that

a_{1} \dots a_{r} = b_{1} \dots b_{s}

with

a_{i}, b_{j} \in A

can only hold when

r = s

?Similarly, can one always find a set

A \subset {1, \dots, N}

with this property of size

\geq (1 - o (1)) N

?A143301 #969Let

Q (x)

count the number of squarefree integers in

[1, x]

. Determine the order of magnitude in the error term in the asymptotic

Q (x) = \frac{6}{π ^{2}} x + E (x) .

A013928

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