A038993 Number of sublattices of index n in generic 6-dimensional lattice.

1, 63, 364, 2667, 3906, 22932, 19608, 97155, 99463, 246078, 177156, 970788, 402234, 1235304, 1421784, 3309747, 1508598, 6266169, 2613660, 10417302, 7137312, 11160828, 6728904, 35364420, 12714681, 25340742, 25095280, 52294536, 21243690, 89572392, 29583456

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

References

• Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
• Index entries for sequences related to sublattices.

Crossrefs

Column 6 of A160870.

Cf. A001001, A038991, A038992, A038994, A038995, A038996, A038997, A038998, A038999.

Programs

Mathematica

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 5}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=6.

Multiplicative with a(p^e) = product (p^(e+k)-1)/(p^k-1), k=1..5.

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4)*zeta(s-5). Dirichlet convolution of A038992 with A000584. - R. J. Mathar, Mar 31 2011

Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^12*zeta(3)*zeta(5)/3061800 = 0.376266... . - Amiram Eldar, Oct 19 2022

Extensions

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Amiram Eldar, Aug 29 2019