

A050999


Sum of squares of odd divisors of n.


25



1, 1, 10, 1, 26, 10, 50, 1, 91, 26, 122, 10, 170, 50, 260, 1, 290, 91, 362, 26, 500, 122, 530, 10, 651, 170, 820, 50, 842, 260, 962, 1, 1220, 290, 1300, 91, 1370, 362, 1700, 26, 1682, 500, 1850, 122, 2366, 530, 2210, 10, 2451, 651, 2900, 170, 2810, 820, 3172, 50, 3620, 842, 3482
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OFFSET

1,3


COMMENTS

Denoted by Delta_2(n) in Glaisher 1907.  Michael Somos, May 17 2013
The sum of squares of even divisors of 2*k = 4*A001157(k), and the sum of squares of even divisors of 2*k1 vanishes, for k >= 1.  Wolfdieter Lang, Jan 07 2017


REFERENCES

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 162 (see p. 4).


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 162 (see p. 4 and p. 8).
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT] (2011), eq. (3.74)
Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences mentioned by Glaisher


FORMULA

Multiplicative with a(p^e) = 1 if p = 2, (p^(2e+2)1)/(p^21) if p > 2. a(n) = 1/2*Sum_{dn} ((1(1)^d)*d^2. a(2n)=sigma_2(2n)4*sigma_2(n), a(2n+1)=sigma_2(2n+1), where sigma_2(n) is sum of squares of divisors of n (A001157). More generally, if b(n, k) is sum of kth powers of odd divisors of n then b(2n, k) = sigma_k(2n)2^k*sigma_k(n), b(2n+1, k) =sigma_k(2n+1). b(n, k) is multiplicative with a(p^e) = 1 if p = 2, (p^(ke+k)1)/(p^k1) if p > 2.  Vladeta Jovovic, Sep 10 2001
G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1(2^k1)*x^m)/(1x^(2*m)).  Vladeta Jovovic, Oct 19 2002
Dirichlet g.f. (12^(2s))*zeta(s)*zeta(s2).  R. J. Mathar, Apr 06 2011
Dirichlet convolution of A001157 with [1,4,0,0,0,0...]. Dirichlet convolution of [1,3,1,3,1,3,..] with A000290. Dirichlet convolution of [1,0,9,0,25,0,49,0,81,...] with A000012 (or A057427).  R. J. Mathar, Jun 28 2011
a(n) = sum(A182469(n,k)^2: k=1..A001227(n)). [Reinhard Zumkeller, May 01 2012]
Sum_{k=1..n} a(k) ~ Zeta(3) * n^3 / 6.  Vaclav Kotesovec, Nov 09 2018


EXAMPLE

x + x^2 + 10*x^3 + x^4 + 26*x^5 + 10*x^6 + 50*x^7 + x^8 + 91*x^9 + 26*x^10 + ...


MATHEMATICA

a[n_] := 1/2*Sum[(1  (1)^d)*d^2, {d, Divisors[n]}]; Table[a[n], {n, 1, 59}] (* JeanFrançois Alcover, Oct 23 2012, from 2nd formula *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d^2, {d, Divisors@n}]] (* Michael Somos, May 17 2013 *)


PROG

(Haskell)
a050999 = sum . map (^ 2) . a182469_row
 Reinhard Zumkeller, May 01 2012
(PARI) a(n)=sumdiv(n, d, if(d%2==1, d^2, 0 ) ); /* Joerg Arndt, Oct 07 2012 */


CROSSREFS

Cf. A051000  A051002, A000593, A001227, A000203, A001157A001160, A013954A013972.
Glaisher's Delta_i (i=0..12): A001227, A000593, A050999, A051000, A051001, A051002, A321810, A321811, A321812, A321813, A321814, A321815, A321816
Sequence in context: A036188 A013617 A243002 * A223450 A221311 A070246
Adjacent sequences: A050996 A050997 A050998 * A051000 A051001 A051002


KEYWORD

nonn,mult


AUTHOR

Eric W. Weisstein


STATUS

approved



