Search: id:A039956
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%I A039956
%S A039956 2,6,10,14,22,26,30,34,38,42,46,58,62,66,70,74,78,82,86,94,102,106,
%T A039956 110,114,118,122,130,134,138,142,146,154,158,166,170,174,178,182,
%U A039956 186,190,194,202,206,210,214,218,222,226,230,238,246,254,258,262
%N A039956 Even square-free numbers.
%C A039956 Sum of even divisors = 2* the sum of odd divisors. - Amarnath Murthy 
               (amarnath_murthy(AT)yahoo.com), Sep 07 2002
%C A039956 Contribution from Daniel Forgues (squid(AT)zensearch.com), May 27 2009: 
               (Start)
%C A039956 a(n) = n * (3/1) * zeta(2) + O(n^(1/2)) = n * (3/1) * (pi^2 / 6) + O(n^(1/
               2))
%C A039956 For any prime p_i, the n_th squarefree number even to p_i (divisible 
               by p_i) is:
%C A039956 n * ((p_i + 1)/1) * zeta(2) + O(n^(1/2)) = n * (p_i + 1)/1) * (pi^2 / 
               6) + O(n^(1/2))
%C A039956 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).
%C A039956 E.g. there are as many even squarefree numbers as there are odd squarefree 
               numbers.
%C A039956 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.
%C A039956 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.
%C A039956 (End)
%D A039956 R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
%H A039956 T. D. Noe, <a href="b039956.txt">Table of n, a(n) for n=1..10000</a>
%F A039956 n such that A092673(n)=+/-2 - Jon Perry (perry(AT)globalnet.co.uk), Mar 
               02 2004
%Y A039956 Cf. A005117, A056911, A039955, A039957.
%Y A039956 Sequence in context: A103747 A000952 A164302 this_sequence A118369 A082816 
               A074105
%Y A039956 Adjacent sequences: A039953 A039954 A039955 this_sequence A039957 A039958 
               A039959
%K A039956 nonn,nice,easy
%O A039956 1,1
%A A039956 R. K. Guy (rkg(AT)cpsc.ucalgary.ca)

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