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A113184 Absolute difference between sum of odd divisors of n and sum of even divisors of n. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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.

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

EXAMPLE

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)

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 *)

PROG

(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)))))}

CROSSREFS

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

Cf. A072558, A284362.

Sequence in context: A196999 A090370 A002129 * A136004 A248864 A134299

Adjacent sequences:  A113181 A113182 A113183 * A113185 A113186 A113187

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Oct 17 2005

EXTENSIONS

Name corrected by Wolfdieter Lang, Jan 07 2017

STATUS

approved

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Last modified July 6 20:41 EDT 2022. Contains 355114 sequences. (Running on oeis4.)