OFFSET
1,1
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.
J. D. Phillips, An analogue of Carmichael numbers, or six for infinity, Journal Articles. 512 (2026). See p. 3.
FORMULA
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
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 *)
PROG
(Magma) [n: n in [2..262 by 2] | IsSquarefree(n)]; // Bruno Berselli, Mar 03 2011
(Haskell)
a039956 n = a039956_list !! (n-1)
a039956_list = filter even a005117_list -- Reinhard Zumkeller, Aug 15 2011
(PARI) is(n)=n%4==2 && issquarefree(n) \\ Charles R Greathouse IV, Sep 13 2013
(Python)
from math import isqrt
from sympy import mobius
def A039956(n):
def f(x): return int(n+x-sum(mobius(k)*(x//k**2+1>>1) for k in range(1, isqrt(x)+1, 2)))
m, k = n, f(n)
while m != k: m, k = k, f(k)
return m<<1 # Chai Wah Wu, Nov 25 2025
CROSSREFS
KEYWORD
nonn,nice,easy,changed
AUTHOR
STATUS
approved
