1,1

Sum of even divisors = 2* the sum of odd divisors. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 07 2002

Contribution from Daniel Forgues (squid(AT)zensearch.com), 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 correspondance, 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)

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

T. D. Noe, Table of n, a(n) for n=1..10000

n such that A092673(n)=+/-2 - Jon Perry (perry(AT)globalnet.co.uk), Mar 02 2004

Cf. A005117, A056911, A039955, A039957.

Adjacent sequences: A039953 A039954 A039955 this_sequence A039957 A039958 A039959

Sequence in context: A103747 A000952 A164302 this_sequence A118369 A082816 A074105

nonn,nice,easy

R. K. Guy (rkg(AT)cpsc.ucalgary.ca)