A113063 - OEIS

%I #31 Dec 28 2023 01:51:26

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

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

%U 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 Associated with theta series of hexagonal net with respect to a node.

%C Denoted by |lambda(n)| on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - _Michael Somos_, Jun 04 2015

%H Antti Karttunen, <a href="/A113063/b113063.txt">Table of n, a(n) for n = 1..16384</a>

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]

%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:1610.07793 [math.NT], 2016.

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

%F a(n) is multiplicative with 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) = A113062(n) unless n=0.

%F G.f.: Sum_{k>0} f(x^k) + f(x^(3*k)) where f(x) := x / (1 + x + x^2). - _Michael Somos_, Jun 04 2015

%F a(n) = |A123477(n)|. - _Michael Somos_, Dec 10 2017

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(9*sqrt(3)) = 0.806133... (A121839 - 1). - _Amiram Eldar_, Dec 28 2023

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

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, 1, 1, -1, -1, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* _Michael Somos_, Jun 04 2015 *)

%t f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[3, e_] := 2; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Sep 05 2023 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, 1, 1, -1, -1, 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, A033687, A113062, A121839, A123477.

%K nonn,easy,mult

%O 1,3

%A _Michael Somos_, Oct 13 2005