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A050999 Sum of squares of odd divisors of n. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Denoted by Delta_2(n) in Glaisher 1907. - Michael Somos, May 17 2013

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

Eric Weisstein's World of Mathematics, Odd Divisor Function

FORMULA

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 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^(ke+k)-1)/(p^k-1) if p > 2. - Vladeta Jovovic, Sep 10 2001

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]

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 *)

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, A001157-A001160, A013954-A013972.

Sequence in context: A040109 A036188 A013617 * A223450 A221311 A070246

Adjacent sequences:  A050996 A050997 A050998 * A051000 A051001 A051002

KEYWORD

nonn,mult

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified July 30 23:15 EDT 2014. Contains 245076 sequences.