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A281785 a(n) is multiplicative with a(2^e) = 1, a(3^e) = -8 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

%I A281785 #23 Oct 24 2023 04:33:52
%S A281785 1,1,-8,1,6,-8,8,1,-8,6,12,-8,14,8,-48,1,18,-8,20,6,-64,12,24,-8,31,
%T A281785 14,-8,8,30,-48,32,1,-96,18,48,-8,38,20,-112,6,42,-64,44,12,-48,24,48,
%U A281785 -8,57,31,-144,14,54,-8,72,8,-160,30,60,-48,62,32,-64,1,84
%N A281785 a(n) is multiplicative with a(2^e) = 1, a(3^e) = -8 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
%C A281785 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A281785 Robert Israel, <a href="/A281785/b281785.txt">Table of n, a(n) for n = 1..10000</a>
%F A281785 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.
%F A281785 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.
%F A281785 3 * a(n) = A281786(n) if n>0. a(2*n) = a(n). a(3*n) = -8 * A186099(n).
%F A281785 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
%e A281785 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 + ...
%p A281785 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]):
%p A281785 map(f, [$1..100]); # _Robert Israel_, Feb 01 2017
%t A281785 a[ n_] := If[ n < 1, 0, If[ Divisible[n, 3], -8, 1] DivisorSigma[ 1, n / (2^IntegerExponent[n, 2] 3^IntegerExponent[n, 3])]];
%t A281785 a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, -8, True, (#^(#2+1) - 1) / (# - 1)] & @@@ FactorInteger@n)];
%o A281785 (PARI) {a(n) = if( n<1, 0, if( n%3, 1, -8) * sigma(n / (2^valuation(n, 2) * 3^valuation(n, 3))))};
%o A281785 (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))))};
%Y A281785 Cf. A186099, A281786.
%K A281785 mult,sign
%O A281785 1,3
%A A281785 _Michael Somos_, Jan 30 2017