A002129 Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n. (Formerly M3236 N1307)

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

1,3

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

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

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A016719 A196999 A090370 * A113184 A136004 A248864

Adjacent sequences: A002126 A002127 A002128 * A002130 A002131 A002132

Keyword

sign,easy,nice,mult

Author

N. J. A. Sloane

Extensions

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

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

Status

approved