

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
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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 nth 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 (onetoone 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 nth even squarefree number is very nearly p_i = 2 times the nth odd squarefree number (which means that the nth even squarefree number is very nearly (p_i + 1) = 3 times the nth squarefree number while the nth odd squarefree number is very nearly (p_i + 1)/ p_i = 3/2 the nth 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 tau2atoms as defined in [Anderson, Frazier] and [Lanterman].  Michel Marcus, May 15 2019


REFERENCES

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1B3.


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), 663705. See pp. 698, 699, 702.
G. J. O. Jameson, Even and odd squarefree numbers, Math. Gazette 94 (2010), 123127
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 !! (n1)
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

R. K. Guy


STATUS

approved



