A038997 Number of sublattices of index n in generic 10-dimensional lattice.

1, 1023, 29524, 698027, 2441406, 30203052, 47079208, 408345795, 653757313, 2497558338, 2593742460, 20608549148, 11488207654, 48162029784, 72080070744, 222984027123, 125999618778, 668793731199, 340614792100, 1704167305962, 1389966536992, 2653398536580, 1883023236984

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 10 of A160870.

Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038998, A038999.

Programs

Mathematica

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 9}]; 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=10.

Multiplicative with a(p^e) = Product_{k=1..9} (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^10, where c = Pi^30*zeta(3)*zeta(5)*zeta(7)*zeta(9) / 4511535509250000 = 0.229259... . - 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