A050999 Sum of squares of odd divisors of n.

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

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*k-1 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), 1-62 (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), 1-62 (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

From Vladeta Jovovic, Sep 10 2001: (Start)

Multiplicative with a(p^e) = 1 if p = 2, (p^(2e+2)-1)/(p^2-1) if p > 2.

a(n) = (1/2)*Sum_{d|n} ((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 the sum of k-th 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^(k*e+k)-1)/(p^k-1) if p > 2. (End)

G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1-(2^k-1)*x^m)/(1-x^(2*m)). - Vladeta Jovovic, Oct 19 2002

Dirichlet g.f. (1-2^(2-s))*zeta(s)*zeta(s-2). - 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

G.f.: Sum_{n >= 1} x^n*(1 + 6*x^(2*n) + x^(4*n))/(1 - x^(2*n))^3. - Peter Bala, Dec 19 2021

Sum_{k=1..n} (-1)^(k+1) * a(k) ~ zeta(3) * n^3 / 8. - Vaclav Kotesovec, Aug 07 2022

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}] (* Jean-François 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 *)
f[p_, e_] := If[p == 2, 1, (p^(2*e + 2) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 22 2020 *)

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 */
(Python)
from sympy import divisor_sigma
def A050999(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 2)) # Chai Wah Wu, Jul 16 2022

Crossrefs

Cf. A051000 - A051002, A000593, A001227, A000203, A001157-A001160, A013954-A013972.

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 A361949

Adjacent sequences: A050996 A050997 A050998 * A051000 A051001 A051002

Keyword

nonn,mult

Author

Eric W. Weisstein

Status

approved