A123477 - OEIS

%I #17 Feb 22 2021 08:12:07

%S 1,0,-2,1,0,0,2,0,-2,0,0,-2,2,0,0,1,0,0,2,0,-4,0,0,0,1,0,-2,2,0,0,2,0,

%T 0,0,0,-2,2,0,-4,0,0,0,2,0,0,0,0,-2,3,0,0,2,0,0,0,0,-4,0,0,0,2,0,-4,1,

%U 0,0,2,0,0,0,0,0,2,0,-2,2,0,0,2,0,-2,0,0,-4,0,0,0,0,0,0,4,0,-4,0,0,0,2,0,0,1,0,0,2,0,0

%N Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - _Michael Somos_, Dec 10 2017

%H Seiichi Manyama, <a href="/A123477/b123477.txt">Table of n, a(n) for n = 1..1000</a>

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv preprint arXiv:1610.07793 [math.NT], 2016.

%F Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].

%F a(n) is multiplicative and a(p^e) = -2 if p = 3 and e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).

%F a(3*n + 2) = 0. a(3*n + 1) = A033687(n), a(3*n) = -2*A002324(n).

%F -3*a(n) = A005928(n) unless n=0. |a(n)| = A113063(n).

%e G.f. = q - 2*q^3 + q^4 + 2*q^7 - 2*q^9 - 2*q^12 + 2*q^13 + q^16 + 2*q^19 + ...

%p A123477 := proc(n)

%p     local a,pe,p,e;

%p     a := 1;

%p     for pe in ifactors(n)[2] do

%p         p := op(1,pe) ;

%p         e := op(2,pe) ;

%p         if modp(p,6) = 1 then

%p             a := a*(e+1) ;

%p         elif modp(p,6) in {2,5} then

%p             a := a*(1+(-1)^e)/2 ;

%p         elif e > 0 then

%p             a := -2*a ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A123477(n),n=1..100) ; # _R. J. Mathar_, Feb 22 2021

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, -3, 1, -1, 3, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* _Michael Somos_, Dec 10 2017 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, p%6==1, e+1, !(e%2))))};

%Y Cf. A002324, A005928, A033687, A113063.

%K sign,mult

%O 1,3

%A _Michael Somos_, Sep 27 2006