A038994 Number of sublattices of index n in generic 7-dimensional lattice.

1, 127, 1093, 10795, 19531, 138811, 137257, 788035, 896260, 2480437, 1948717, 11798935, 5229043, 17431639, 21347383, 53743987, 25646167, 113825020, 49659541, 210837145, 150021901, 247487059, 154764793, 861322255, 317886556, 664088461, 678468820, 1481689315

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 7 of A160870.

Cf. A001001, A038991, A038992, A038993, A038995, A038996, A038997, A038998, A038999.

Programs

Mathematica

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 6}]; 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=7.

Multiplicative with a(p^e) = Product_{k=1..6} (p^(e+k)-1)/(p^k-1).

Sum_{k=1..n} a(k) ~ c * n^7, where c = Pi^12*zeta(3)*zeta(5)*zeta(7)/3572100 = 0.325206... . - Amiram Eldar, Oct 19 2022

Extensions

More terms from Amiram Eldar, Aug 29 2019