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A039956 Even squarefree numbers. 20
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

REFERENCES

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

LINKS

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

G. J. O. Jameson, Even and odd square-free numbers, Math. Gazette 94 (2010), 123-127

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

R. K. Guy

STATUS

approved

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Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.