A038992 Number of sublattices of index n in generic 5-dimensional lattice.

1, 31, 121, 651, 781, 3751, 2801, 11811, 11011, 24211, 16105, 78771, 30941, 86831, 94501, 200787, 88741, 341341, 137561, 508431, 338921, 499255, 292561, 1429131, 508431, 959171, 925771, 1823451, 732541, 2929531, 954305, 3309747, 1948705, 2750971, 2187581, 7168161, 1926221

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 (terms 1..5000 from G. C. Greubel)
• M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
• Index entries for sequences related to sublattices.

Crossrefs

Column 5 of A160870.

Cf. A001001, A038991, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

Programs

Mathematica

a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #*DivisorSum[#, # &] &] &] &]; Array[a, 50] (* Jean-François Alcover, Dec 02 2015, after Joerg Arndt *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 4}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

PARI

a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z * sumdiv(z,w, w ) ) ) ); /* Joerg Arndt, Oct 07 2012 */

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=5.

Multiplicative with a(p^e) = Product_{k=1..4} (p^(e+k)-1)/(p^k-1).

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4). Dirichlet convolution of A038991 with A000583. - R. J. Mathar, Mar 31 2011

Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^6*zeta(3)*zeta(5)/2700 = 0.443822... . - Amiram Eldar, Oct 19 2022

Extensions

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Joerg Arndt, Oct 07 2012