A113184 Absolute difference between sum of odd divisors of n and 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, 84, 48, 68, 90, 96, 48, 72

Offset: 1

Author

Michael Somos, Oct 17 2005

Keywords

• nonn
• mult

Comments

The generating function equals 1/8 at q = Lambda = 0.1076539192... (A072558) the "One-Ninth" constant. - Michael Somos, Jul 21 2006

Absolute value of A002129. - John W. Layman, Sep 27 2012

The Möbius transform is 1, 0, 3, 4, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 16, 17, 0, 19, 20, 21, 0, 23, 24, 25, 0, 27, ... - R. J. Mathar, Jan 08 2013

Examples

			From _Peter Bala_, Dec 11 2020: (Start)
n = 15: n is a triangular number, so e(n) = (-1)^(n+1)*n = 15 and a(15) = 15 + a(14) + a(12) - a(9) - a(5) =  15 + 8 + 20 - 13 - 6 = 24;
n = 16: n is a not triangular number, so e(n) = 0 and a(16) = a(15) + a(13) - a(10) - a(6) + a(1) =  24 + 14 - 6 - 4 + 1 = 29. (End)
		

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, One-Ninth Constant.

Crossrefs

Cf. A002129(n) = -(-1)^n a(n).

Cf. A072558, A284362.

Programs

Mathematica

f[n_]:=Module[{dn=Divisors[n],odn,edn},odn=Select[dn,OddQ];edn=Select[dn,EvenQ];Abs[Total[odn]-Total[edn]]]
f/@Range[80]  (* Harvey P. Dale, Feb 25 2011 *)
max = 80; s = (1/x)*Sum[k*x^k/(1 - (-x)^k), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 04 2015 *)
f[p_, e_] := If[p == 2, 2^(e + 1) - 3, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

PARI

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

PARI

{a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, 2^(e+1)-3, (p^(e+1)-1)/(p-1)))))}

Formula

Multiplicative with a(2^e) = 2^(e+1)-3 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p>2.

G.f.: Sum_{k>0} -(-x)^k/(1+(-x)^k)^2 = Sum_{k>0} k*x^k/(1-(-x)^k).

Expansion of (1-(2/Pi)^2(2E(k)-K(k))K(k))/8 in powers of nome q where E(k) and K(k) are complete elliptic integrals and q=exp(-Pi*K(k')/K(k)). - Michael Somos, Jul 21 2006

Bisection: a(2*k-1) = A000203(2*k-1), a(2*k) = A146076(2*k) - A000593(2*k), k >= 1. See the Hardy reference where a(n) = sigma^*1(n). - _Wolfdieter Lang, Jan 07 2017

From Peter Bala, Dec 11 2020: (Start)

a(n) = Sum_{d | n, d != 2 (mod 4)} d.

O.g.f.: Sum_{k >= 1, k != 2 (mod 4)} k*x^k/(1 - x^k). Cf. A284362.

Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-3) - a(n-6) - a(n-10) + + - -, where [1, 3, 6, 10, ...] is the sequence of triangular numbers A000217, and e(n) = (-1)^(n+1)*n if n is a triangular number; otherwise e(n) = 0. Examples of this recurrence are given below. (End)

Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-1) * (1+2^(3-3*s)) / (1+2^(1-s)). - Werner Schulte, Jan 23 2021

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 16. - Vaclav Kotesovec, Aug 20 2021

Extensions

Name corrected by Wolfdieter Lang, Jan 07 2017