A002129 Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.

1, -1, 4, -5, 6, -4, 8, -13, 13, -6, 12, -20, 14, -8, 24, -29, 18, -13, 20, -30, 32, -12, 24, -52, 31, -14, 40, -40, 30, -24, 32, -61, 48, -18, 48, -65, 38, -20, 56, -78, 42, -32, 44, -60, 78, -24, 48, -116, 57, -31, 72, -70, 54, -40, 72, -104, 80, -30, 60, -120, 62, -32, 104, -125

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice
• mult

Comments

Glaisher calls this zeta(n) or zeta_1(n). - N. J. A. Sloane, Nov 24 2018

Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n)^2 = Sum_{n >= 1} (-1)^(n-1)*n*x^n/(1-x^n).

Unsigned sequence is A113184. - Peter Bala, Dec 14 2020

Examples

			a(28) = 40 because the sum of the even divisors of 28 (2, 4, 14 and 28) = 48 and the sum of the odd divisors of 28 (1 and 7) = 8, their absolute difference being 40.
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 3rd formula.
• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Steven R. Finch, The "One-Ninth" Constant [Broken link]
• Steven R. Finch, The "One-Ninth" Constant [From the Wayback machine]
• 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).
• Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
• Index entries for sequences mentioned by Glaisher

Crossrefs

A diagonal of A060044.

a(2^n) = -A036563(n+1). a(3^n) = A003462(n+1).

First differences of -A024919(n).

Cf. A010054, A113184.

Programs

Maple

A002129 := proc(n) -add((-1)^d*d,d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 05 2011

Mathematica

f[n_] := Block[{c = Divisors@ n}, Plus @@ Select[c, EvenQ] - Plus @@ Select[c, OddQ]]; Array[f, 64] (* Robert G. Wilson v, Mar 04 2011 *)
a[n_] := DivisorSum[n, -(-1)^#*#&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := If[p == 2, 3 - 2^(e + 1), (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]);  Array[a, 64] (* Amiram Eldar, Jul 20 2019 *)

PARI

a(n)=if(n<1,0,-sumdiv(n,d,(-1)^d*d))

PARI

{a(n)=n*polcoeff(log(sum(k=0,(sqrtint(8*n+1)-1)\2,x^(k*(k+1)/2))+x*O(x^n)),n)} \\ Paul D. Hanna, Jun 28 2008

Formula

Multiplicative with a(p^e) = 3-2^(e+1) if p = 2; (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Sep 01 2001

G.f.: Sum_{n>=1} n*x^n*(1-3*x^n)/(1-x^(2*n)). - Vladeta Jovovic, Oct 15 2002

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log[ Sum_{n>=0} x^(n(n+1)/2) ], the log of the g.f. of A010054. - Paul D. Hanna, Jun 28 2008

Dirichlet g.f. zeta(s)*zeta(s-1)*(1-4/2^s). Dirichlet convolution of A000203 and the quasi-finite (1,-4,0,0,0,...). - R. J. Mathar, Mar 04 2011

a(n) = A000593(n)-A146076(n). - R. J. Mathar, Mar 05 2011

a(n) = Sum_{j = 1..n} Sum_{k = 1..j} (-1)^(j+1)*cos(2*k*n*Pi/j). - Peter Bala, Aug 24 2022

G.f.: Sum_{n>=1} n*(-x)^(n-1)/(1-x^n). - Mamuka Jibladze, Jun 03 2025

Extensions

Better description and more terms from Robert G. Wilson v, Dec 14 2000

More terms from N. J. A. Sloane, Mar 19 2001