A038999 Sublattices of index n in generic 12-dimensional lattice.

1, 4095, 265720, 11180715, 61035156, 1088123400, 2306881200, 26167664835, 52955405230, 249938963820, 313842837672, 2970939589800, 1941507093540, 9446678514000, 16218261652320, 57162391576563, 36413889826860, 216852384416850, 122961939948120, 682416684216540

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

Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998.

Programs

Mathematica

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 11}]; 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=12.

Multiplicative with a(p^e) = Product_{k=1..11} (p^(e+k)-1)/(p^k-1).

Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k).

Sum_{k=1..n} a(k) ~ c * n^12, where c = Pi^42*zeta(3)*zeta(5)*zeta(7)*zeta(9)*zeta(11)/3456808210410967912500000 = 0.191191... . - Amiram Eldar, Oct 19 2022

Extensions

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

More terms from Amiram Eldar, Aug 29 2019