OFFSET
1,3
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
Expansion of (a(x) * b(x^2) + a(x^2) * b(x) - 2) / 3 in powers of x where a(), b() are cubic AGM functions.
Expansion of (3 * b(x^3) * b(x^6) - b(x) * b(x^2) - 2) / 3 in powers of x where b() is a cubic AGM function.
Dirichlet g.f.: zeta(s) * zeta(s-1) * (1-2^(1-s)) * (1-3^(1-s)) * (1-3^(2-s)). - Amiram Eldar, Oct 24 2023
EXAMPLE
G.f. = x + x^2 - 8*x^3 + x^4 + 6*x^5 - 8*x^6 + 8*x^7 + x^8 - 8*x^9 + 6*x^10 + ...
MAPLE
f:= n -> mul(piecewise( t[1] = 2, 1, t[1] = 3, -8, (t[1]^(t[2]+1)-1)/(t[1]-1)), t = ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Feb 01 2017
MATHEMATICA
a[ n_] := If[ n < 1, 0, If[ Divisible[n, 3], -8, 1] DivisorSigma[ 1, n / (2^IntegerExponent[n, 2] 3^IntegerExponent[n, 3])]];
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, -8, True, (#^(#2+1) - 1) / (# - 1)] & @@@ FactorInteger@n)];
PROG
(PARI) {a(n) = if( n<1, 0, if( n%3, 1, -8) * sigma(n / (2^valuation(n, 2) * 3^valuation(n, 3))))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*(d%2)) - if( n%3==0, 12 * sumdiv(n/3, d, d*(d%2))) + if( n%9==0, 27 * sumdiv(n/9, d, d*(d%2))))};
CROSSREFS
KEYWORD
mult,sign
AUTHOR
Michael Somos, Jan 30 2017
STATUS
approved