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A039956 Even squarefree numbers. 45

%I #87 Sep 26 2023 02:09:24

%S 2,6,10,14,22,26,30,34,38,42,46,58,62,66,70,74,78,82,86,94,102,106,

%T 110,114,118,122,130,134,138,142,146,154,158,166,170,174,178,182,186,

%U 190,194,202,206,210,214,218,222,226,230,238,246,254,258,262

%N Even squarefree numbers.

%C Sum of even divisors = 2 * the sum of odd divisors. - _Amarnath Murthy_, Sep 07 2002

%C From _Daniel Forgues_, May 27 2009: (Start)

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

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

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

%C 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).

%C E.g., there are as many even squarefree numbers as there are odd squarefree numbers.

%C 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 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 (End)

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

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

%H T. D. Noe, <a href="/A039956/b039956.txt">Table of n, a(n) for n = 1..10000</a>

%H D. D. Anderson and Andrea M. Frazier, <a href="https://doi.org/10.1216/RMJ-2011-41-3-663">On a general theory of factorization in integral domains</a>, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.

%H G. J. O. Jameson, <a href="https://www.jstor.org/stable/27821897">Even and odd square-free numbers</a>, Math. Gazette 94 (2010), 123-127; <a href="http://www.maths.lancs.ac.uk/~jameson/sqf.pdf">Author's copy</a>.

%H James Lanterman, <a href="https://arxiv.org/abs/1210.2991">Irreducibles in the Integers modulo n</a>, arXiv:1210.2991 [math.NT], 2012.

%F Numbers k such that A092673(k) = +- 2. - _Jon Perry_, Mar 02 2004

%F Sum_{n>=1} 1/a(n)^s = zeta(s)/((1+2^s)*zeta(2*s)). - _Enrique PĂ©rez Herrero_, Sep 15 2012 [corrected by _Amiram Eldar_, Sep 26 2023]

%F a(n) = 2*A056911(n). - _Robert Israel_, Dec 23 2015

%F a(n) = 2*(1+2*A264387(n)), n >= 1. - _Wolfdieter Lang_, Dec 24 2015

%p select(numtheory:-issqrfree,[seq(i,i=2..1000,4)]); # _Robert Israel_, Dec 23 2015

%t Select[Range[2,270,2],SquareFreeQ] (* _Harvey P. Dale_, Jul 23 2011 *)

%o (Magma) [n: n in [2..262 by 2] | IsSquarefree(n)]; // _Bruno Berselli_, Mar 03 2011

%o (Haskell)

%o a039956 n = a039956_list !! (n-1)

%o a039956_list = filter even a005117_list -- _Reinhard Zumkeller_, Aug 15 2011

%o (PARI) is(n)=n%4==2 && issquarefree(n) \\ _Charles R Greathouse IV_, Sep 13 2013

%Y Subsequence of A005117.

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

%K nonn,nice,easy

%O 1,1

%A _R. K. Guy_

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)