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
KEYWORD
nonn,mult,changed
AUTHOR
Michael Somos, Oct 17 2005
EXTENSIONS
Name corrected by Wolfdieter Lang, Jan 07 2017
STATUS
approved