A115607 Sum of odd divisors of n times (-1)^(n+1).

1, -1, 4, -1, 6, -4, 8, -1, 13, -6, 12, -4, 14, -8, 24, -1, 18, -13, 20, -6, 32, -12, 24, -4, 31, -14, 40, -8, 30, -24, 32, -1, 48, -18, 48, -13, 38, -20, 56, -6, 42, -32, 44, -12, 78, -24, 48, -4, 57, -31, 72, -14, 54, -40, 72, -8, 80, -30, 60, -24, 62, -32, 104, -1, 84, -48, 68, -18, 96, -48, 72, -13, 74, -38, 124

Offset

1,3

References

C. G. J. Jacobi, Gesammelte Werke, Vol. 2, 1969, Chelsea, NY, p. 179, eq (6.).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

Moebius transform is [ 1, -2, 3, 0, 5, -6, 7, 0, 9, -10, 11, 0, ...].

a(n) is multiplicative with a(2^e) = -1 if e > 0, a(p^e) = (p^(e+1) - 1) / (p-1) if p > 2.

G.f.: (theta_2(q)^4 - theta_4(q)^4 + 1) / 24.

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

G.f.: Sum_{k>0} (2*k - 1) * x^(2*k - 1) / (1 + x^(2*k - 1)) = Sum_{k>0} x^k / (1 - x^k)^2 - 4 * x^(4*k - 2) / (1 - x^(4*k - 2))^2.

Empirical: Sum_{n>=1} exp(-Pi)^n*a(n) = 1/24. - Simon Plouffe, Feb 20 2011

Dirichlet g.f. (1-2^(1-s))^2*zeta(s)*zeta(s-1). - R. J. Mathar, Apr 06 2011

a(n) = -(-1)^n * A000593(n). -24 * a(n) = A103640(n) if n > 0.

G.f.: Sum_{k>0} -(-1)^k * x^k * (1 + x^(2*k)) / (1 - x^(2*k))^2. - Michael Somos, Jun 26 2012

L.g.f.: -log(Product_{ k>0 } (1+(-x)^k)) = Sum_{ n>=0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 04 2016

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/48 = 0.205616... . - Amiram Eldar, Nov 12 2022

Example

x - x^2 + 4*x^3 - x^4 + 6*x^5 - 4*x^6 + 8*x^7 - x^8 + 13*x^9 - 6*x^10 + 12*x^11 + ...

Maple

with(numtheory):
a:= n-> add(`if`(d::odd, -d, 0), d=divisors(n))*(-1)^n:
seq(a(n), n=1..75);  # Alois P. Heinz, Apr 07 2017

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) n / # &]] (* Michael Somos, Jun 17 2012 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 2, 0, q]^4 - EllipticTheta[ 4, 0, q]^4 + 1) / 24, {q, 0, n}] (* Michael Somos, Jun 26 2012 *)
Table[CoefficientList[Series[-Log[QPochhammer[-1, -x]/2], {x, 0, 60}], x][[n + 1]] n, {n, 1, 60}] (* Benedict W. J. Irwin, Jul 04 2016 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(n + d) * n/d))}
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, d%2*d))}
(Python)
from sympy import divisors
def a(n): return 0 if n<1 else -(-1)**n * sum([d%2*d for d in divisors(n)]) # Indranil Ghosh, Apr 09 2017

Crossrefs

Cf. A000593, A103640.

Sequence in context: A336113 A098986 A000593 * A318876 A330356 A329428

Adjacent sequences: A115604 A115605 A115606 * A115608 A115609 A115610

Keyword

sign,look,mult

Author

Michael Somos, Jan 26 2006

Status

approved