A123314 -id:A123314 - cp's OEIS frontend

A056911 Odd squarefree numbers.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149, 151

Offset: 1

James Sellers, Jul 07 2000

From Daniel Forgues, May 27 2009: (Start)
For any prime p, there are as many squarefree numbers having p as a factor as squarefree numbers not having p as a factor amongst all the squarefree numbers (one-to-one correspondence, both cardinality aleph_0).
E.g. there are as many even squarefree numbers as there are odd squarefree numbers.
For any prime p, the density of squarefree numbers having p as a factor is 1/p of the density of squarefree numbers not having p as a factor.
E.g. the density of even squarefree numbers is 1/p = 1/2 of the density of odd squarefree numbers (which means that 1/(p + 1) = 1/3 of the squarefree numbers are even and p/(p + 1) = 2/3 are odd). As a consequence the n-th even squarefree number is very nearly p = 2 times the n-th odd squarefree number (which means that the n-th even squarefree number is very nearly (p + 1) = 3 times the n-th squarefree number while the n-th odd squarefree number is very nearly (p + 1)/p = 3/2 the n-th squarefree number).
For any prime p, the n-th squarefree number not divisible by p is: n * (1 + 1/p) * zeta(2) + O(n^(1/2)) = n * (1 + 1/p) * (Pi^2 / 6) + O(n^(1/2)) (End)

			The exponents in the prime factorization of 15 are all equal to 1, so 15 appears here. The number 75 does not appear in this sequence, as it is divisible by the square number 25.

Zak Seidov, Table of n, a(n) for n = 1..12000
G. J. O. Jameson, Even and odd square-free numbers, Math. Gazette 94 (2010), 123-127; Author's copy.

Subsequence of A005117 and A036537.
Equals A039956/2.
Cf. A238711 (subsequence).

a056911 n = a056911_list !! (n-1)
a056911_list = filter ((== 1) . a008966) [1,3..]
-- Reinhard Zumkeller, Aug 27 2011

[n: n in [1..151 by 2] | IsSquarefree(n)]; // Bruno Berselli, Mar 03 2011

Select[Range[1,151,2],SquareFreeQ] (* Ant King, Mar 17 2013 *)

is(n)=n%2 && issquarefree(n) \\ Charles R Greathouse IV, Mar 26 2013

list(lim)=my(v=List()); forsquarefree(k=1,lim\1, if(k[1]%2, listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2025

A123314(A100112(a(n))) > 0. - Reinhard Zumkeller, Sep 25 2006
a(n) = n * (3/2) * zeta(2) + O(n^(1/2)) = n * (Pi^2 / 4) + O(n^(1/2)). - Daniel Forgues, May 27 2009
A008474(a(n)) * A000035(a(n)) = 1. - Reinhard Zumkeller, Aug 27 2011
Sum_{n>=1} 1/a(n)^s = ((2^s)* zeta(s))/((1+2^s)*zeta(2*s)). - Enrique Pérez Herrero, Sep 15 2012 [corrected by Amiram Eldar, Sep 26 2023]

A166607 Numbers k such that Sum_{i=1..k} i^7 divides Product_{i=1..k} i^7.

Original entry on oeis.org

1, 733, 1637, 2096, 2367, 4231, 5674, 5839, 7585, 8344, 13719, 13753, 14983, 15151, 15197, 15257, 15757, 16595, 17305, 18791, 20701, 21442, 23652, 23738, 24519, 24789, 25474, 25916, 25933, 27474, 27487, 29185, 31455, 32846, 32950, 33421

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(7) = 733.

Cf. A123314, A166602, A060462, A166604, A166605, A166606, A166608, A166609, A166610.

With[{c=Range[40000]^7},Flatten[Position[#[[1]]/#[[2]]&/@Thread[ {Rest[ FoldList[ Times,1,c]],Accumulate[c]}],?IntegerQ]]] (* _Harvey P. Dale, Nov 16 2014 *)

More terms from Max Alekseyev, Sep 30 2010

A123313 Maximum of greatest common divisors of pairs of distinct squarefree numbers not greater than the n-th squarefree number; a(1)=1.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 5, 5, 7, 7, 7, 7, 7, 11, 11, 13, 13, 15, 15, 15, 17, 17, 17, 19, 19, 19, 21, 21, 23, 23, 23, 23, 23, 23, 29, 29, 29, 31, 31, 33, 33, 33, 35, 35, 35, 37, 37, 39, 39, 41, 41, 41, 43, 43, 43, 43, 43, 47, 47, 47, 47, 51, 51, 51, 53, 53, 53, 55, 55, 55, 57, 57, 59

Offset: 1

Reinhard Zumkeller, Sep 25 2006

nonn

A123314(n) = #{k: a(k) = A005117(n)}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.
Index entries for sequences related to GCD's.

Cf. A005117, A123314.

seq[max_] := Module[{sqf = Select[Range[max], SquareFreeQ], s, m}, m = Length[sqf]; s = Table[0, {m}]; s[[1]] = 1; Do[s[[k]] = Max[s[[k - 1]], Max[GCD[sqf[[k]], Take[sqf, k - 1]]]], {k, 2, m}]; s]; seq[120] (* Amiram Eldar, Feb 13 2021 *)

mygcd(x, y) = if (x==y, 1, gcd(x,y));
sqf(n) = {my(v = List(), k=1); until (#v == n, if (issquarefree(k), listput(v, k)); k++); v;}
a(n) = vecmax(setbinop(mygcd, Vec(sqf(n)))); \\ Michel Marcus, Feb 13 2021

a(n) = Max(a(n-1), Max(GCD(A005117(n), A005117(k)): 1<=k

Name corrected by Amiram Eldar, Feb 13 2021

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A056911 Odd squarefree numbers.

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A166607 Numbers k such that Sum_{i=1..k} i^7 divides Product_{i=1..k} i^7.

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A123313 Maximum of greatest common divisors of pairs of distinct squarefree numbers not greater than the n-th squarefree number; a(1)=1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions