A038996 Number of sublattices of index n in generic 9-dimensional lattice.

1, 511, 9841, 174251, 488281, 5028751, 6725601, 50955971, 72636421, 249511591, 235794769, 1714804091, 883708281, 3436782111, 4805173321, 13910980083, 7411742281, 37117211131, 17927094321, 85083452531, 66186639441, 120491126959, 81870575521, 501457710611, 198682027181

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 9 of A160870.

Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038997, A038998, A038999.

Programs

Mathematica

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 8}]; 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=9.

Multiplicative with a(p^e) = Product_{k=1..8} (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^9, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)*zeta(9) / 38578680000 = 0.254479... . - Amiram Eldar, Oct 19 2022

Extensions

Offset changed to 1 by R. J. Mathar, Apr 01 2011

More terms from Amiram Eldar, Aug 29 2019