A050469 a(n) = Sum_{ d divides n, n/d=1 mod 4} d - Sum_{ d divides n, n/d=3 mod 4} d.

1, 2, 2, 4, 6, 4, 6, 8, 7, 12, 10, 8, 14, 12, 12, 16, 18, 14, 18, 24, 12, 20, 22, 16, 31, 28, 20, 24, 30, 24, 30, 32, 20, 36, 36, 28, 38, 36, 28, 48, 42, 24, 42, 40, 42, 44, 46, 32, 43, 62, 36, 56, 54, 40, 60, 48, 36, 60, 58, 48, 62, 60, 42, 64, 84, 40

Offset

1,2

Comments

Multiplicative with a(p^e)=p^e if p=2, (p^(e+1)-1)/(p-1) if p==1 (mod 4), else (p^(e+1)+(-1)^e)/(p+1). - Michael Somos, May 02 2005

Multiplicative because it is the Dirichlet convolution of A000027 = n and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

Links

Seiichi Manyama, 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).

Index entries for sequences mentioned by Glaisher.

Formula

G.f.: Sum_{n>=1} n*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002

L.g.f.: Sum_{k>=1} arctan(x^k). - Ilya Gutkovskiy, Dec 16 2019

O.g.f.: Sum_{n >= 1} (-1)^(n+1) * x^(2*n-1)/(1 - x^(2*n-1))^2. - Peter Bala, Jan 04 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 4)} 1/(1-1/p^2) * Product_{primes p == 3 (mod 4)} 1/(1+1/p^2) = (1/2) * A175647 / A243381 = A006752/2 = 0.4579827970... . - Amiram Eldar, Nov 06 2022, Nov 05 2023

Mathematica

max = 70; s = Sum[n*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 02 2015 *)
f[p_, e_] := Which[p == 2, p^e, Mod[p, 4] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 4] == 3, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 06 2022 *)

Prog

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d*((n/d%4==1)-(n/d%4==3))))
(PARI) {a(n)=local(A, p, e); if(n<2, n==1, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, p^e, if(p%4==1, (p^(e+1)-1)/(p-1), (p^(e+1)+(-1)^e)/(p+1)))))) } /* Michael Somos, May 02 2005 */
(PARI) a(n)=if(n<1, 0, polcoeff(sum(k=1, n, k*x^k/(1+x^(2*k)), x*O(x^n)), n))

Crossrefs

Cf. A006752, A050470, A050471, A050468, A175647, A243381.

Sequence in context: A278973 A073256 A166362 * A350229 A085730 A232065

Adjacent sequences: A050466 A050467 A050468 * A050470 A050471 A050472

Keyword

nonn,mult

Author

N. J. A. Sloane, Dec 23 1999

Status

approved