A038995 Number of sublattices of index n in generic 8-dimensional lattice.

1, 255, 3280, 43435, 97656, 836400, 960800, 6347715, 8069620, 24902280, 21435888, 142466800, 67977560, 245004000, 320311680, 866251507, 435984840, 2057753100, 943531280, 4241688360, 3151424000, 5466151440, 3559590240, 20820505200, 7947261556, 17334277800, 18326727760

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
• Index entries for sequences related to sublattices

Crossrefs

Column 8 of A160870.

Cf. A001001, A038991, A038992, A038993, A038994, A038996, A038997, A038998, A038999.

Programs

Mathematica

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 7}]; 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=8.

Multiplicative with a(p^e) = Product_{k=1..7} (p^(e+k)-1)/(p^k-1).

Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011

Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)/43401015000 = 0.285716... . - Amiram Eldar, Oct 19 2022

Extensions

Offset set to 1 by R. J. Mathar, Mar 01 2011

More terms from Amiram Eldar, Aug 29 2019