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

1, 7, 13, 35, 31, 91, 57, 154, 130, 217, 133, 455, 183, 399, 403, 644, 307, 910, 381, 1085, 741, 931, 553, 2002, 806, 1281, 1209, 1995, 871, 2821, 993, 2632, 1729, 2149, 1767, 4550, 1407, 2667, 2379, 4774, 1723, 5187, 1893, 4655, 4030, 3871

Offset: 1

Author

N. J. A. Sloane, May 11 2001

Keywords

• nonn
• easy
• mult

Comments

These sublattices are in 1-1 correspondence with matrices

[a b d]

[0 c e]

[0 0 f]

with acf = n, b = 0..c-1, d = 0..f-1, e = 0..f-1, gcd(a,b,c,d,e,f) = 1.

From Álvar Ibeas, Oct 30 2015: (Start)

a(n) is the number of 2-generated subgroups of Z^3 with order n.

The Dirichlet convolution of a(n) with A010057 gives A001001.

Dirichlet convolution of A254981 and A000290.

(End)

Links

• Álvar Ibeas, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sublattices.

Crossrefs

Cf. A001001, with which it agrees unless n is divisible by a cube.

Cf. A000290, A010057, A254981.

Programs

Mathematica

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 *)

Formula

From Álvar Ibeas, Oct 30 2015: (Start)

a(n) = Sum_{d^3 | n} mu(d) * A001001(n/d^3).

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) / zeta(3s). (End)

Sum_{k=1..n} a(k) ~ Pi^2 * zeta(3) * n^3 / (18*zeta(9)). - Vaclav Kotesovec, Feb 01 2019

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