A039956 Even squarefree numbers.

2, 6, 10, 14, 22, 26, 30, 34, 38, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 202, 206, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262

Offset: 1

Author

R. K. Guy

Keywords

• nonn
• nice
• easy

Comments

Sum of even divisors = 2 * the sum of odd divisors. - Amarnath Murthy, Sep 07 2002

From Daniel Forgues, May 27 2009: (Start)

a(n) = n * (3/1) * zeta(2) + O(n^(1/2)) = n * (3/1) * (Pi^2 / 6) + O(n^(1/2)).

For any prime p_i, the n-th squarefree number even to p_i (divisible by p_i) is:

n * ((p_i + 1)/1) * zeta(2) + O(n^(1/2)) = n * ((p_i + 1)/1) * (Pi^2 / 6) + O(n^(1/2)).

For any prime p_i, there are as many squarefree numbers having p_i as a factor as squarefree numbers not having p_i 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_i, the density of squarefree numbers having p_i as a factor is 1/p_i of the density of squarefree numbers not having p_i as a factor.

E.g., the density of even squarefree numbers is 1/p_i = 1/2 of the density of odd squarefree numbers (which means that 1/(p_i + 1) = 1/3 of the squarefree numbers are even and p_i/(p_i + 1) = 2/3 are odd) and as a consequence the n-th even squarefree number is very nearly p_i = 2 times the n-th odd squarefree number (which means that the n-th even squarefree number is very nearly (p_i + 1) = 3 times the n-th squarefree number while the n-th odd squarefree number is very nearly (p_i + 1)/ p_i = 3/2 the n-th squarefree number).

(End)

Apart from first term, these are the tau2-atoms as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019

References

• Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
• G. J. O. Jameson, Even and odd square-free numbers, Math. Gazette 94 (2010), 123-127; Author's copy.
• James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.

Crossrefs

Subsequence of A005117.

Cf. A002808, A056911, A039955, A039957, A056911, A092673, A264387.

Programs

Haskell

a039956 n = a039956_list !! (n-1)
a039956_list = filter even a005117_list  -- Reinhard Zumkeller, Aug 15 2011

Magma

[n: n in [2..262 by 2] | IsSquarefree(n)];  // Bruno Berselli, Mar 03 2011

Maple

select(numtheory:-issqrfree,[seq(i,i=2..1000,4)]); # Robert Israel, Dec 23 2015

Mathematica

Select[Range[2,270,2],SquareFreeQ] (* Harvey P. Dale, Jul 23 2011 *)

PARI

is(n)=n%4==2 && issquarefree(n) \\ Charles R Greathouse IV, Sep 13 2013

Formula

Numbers k such that A092673(k) = +- 2. - Jon Perry, Mar 02 2004

Sum_{n>=1} 1/a(n)^s = zeta(s)/((1+2^s)*zeta(2*s)). - Enrique Pérez Herrero, Sep 15 2012 [corrected by Amiram Eldar, Sep 26 2023]

a(n) = 2*A056911(n). - Robert Israel, Dec 23 2015

a(n) = 2*(1+2*A264387(n)), n >= 1. - Wolfdieter Lang, Dec 24 2015