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 A039956 Even squarefree numbers. 29
 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 (list; graph; refs; listen; history; text; internal format)
 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) Sum_{n>=1} a(n)/n^s = zeta(s)/((1+2^s)*zeta(2*s)). - Enrique Pérez Herrero, Sep 15 2012 Apart from first term, these are the tau2-atoms as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019 REFERENCES R. 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 James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012. FORMULA n such that A092673(n) = +- 2. - Jon Perry, Mar 02 2004 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 CROSSREFS Cf. A005117, A056911, A039955, A039957. Sequence in context: A284678 A185548 A239221 * A197930 A192109 A216090 Adjacent sequences:  A039953 A039954 A039955 * A039957 A039958 A039959 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)