cp's OEIS Frontend

A060983 Number of primitive sublattices of index n in generic 3-dimensional lattice.

%I A060983 #26 Aug 27 2023 04:35:09
%S A060983 1,7,13,35,31,91,57,154,130,217,133,455,183,399,403,644,307,910,381,
%T A060983 1085,741,931,553,2002,806,1281,1209,1995,871,2821,993,2632,1729,2149,
%U A060983 1767,4550,1407,2667,2379,4774,1723,5187,1893,4655,4030,3871
%N A060983 Number of primitive sublattices of index n in generic 3-dimensional lattice.
%C A060983 These sublattices are in 1-1 correspondence with matrices
%C A060983   [a b d]
%C A060983   [0 c e]
%C A060983   [0 0 f]
%C A060983 with acf = n, b = 0..c-1, d = 0..f-1, e = 0..f-1, gcd(a,b,c,d,e,f) = 1.
%C A060983 From _Álvar Ibeas_, Oct 30 2015: (Start)
%C A060983 a(n) is the number of 2-generated subgroups of Z^3 with order n.
%C A060983 The Dirichlet convolution of a(n) with A010057 gives A001001.
%C A060983 Dirichlet convolution of A254981 and A000290.
%C A060983 (End)
%H A060983 Álvar Ibeas, <a href="/A060983/b060983.txt">Table of n, a(n) for n = 1..10000</a>
%H A060983 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>.
%F A060983 From _Álvar Ibeas_, Oct 30 2015: (Start)
%F A060983 a(n) = Sum_{d^3 | n} mu(d) * A001001(n/d^3).
%F A060983 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) / zeta(3s). (End)
%F A060983 Sum_{k=1..n} a(k) ~ Pi^2 * zeta(3) * n^3 / (18*zeta(9)). - _Vaclav Kotesovec_, Feb 01 2019
%F A060983 Multiplicative with a(p) = p^2+p+1, and a(p^e) = p^(e-2)*(p^e + (p^(e-1)-1)/(p-1)) for e >= 2. - _Amiram Eldar_, Aug 27 2023
%t A060983 f[p_, e_] := (p^2 + p + 1) * If[e == 1, 1, p^(e - 2)*(p^e + (p^(e - 1) - 1)/(p - 1))]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 27 2023 *)
%Y A060983 Cf. A001001, with which it agrees unless n is divisible by a cube.
%Y A060983 Cf. A000290, A010057, A254981.
%K A060983 nonn,easy,mult
%O A060983 1,2
%A A060983 _N. J. A. Sloane_, May 11 2001