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

1,2

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.

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

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 *)

Prog

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

Crossrefs

Column 5 of A160870.

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

Sequence in context: A158558 A160893 A202994 * A068021 A131992 A042884

Adjacent sequences: A038989 A038990 A038991 * A038993 A038994 A038995

Keyword

nonn,mult

Author

N. J. A. Sloane

Extensions

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

More terms from Joerg Arndt, Oct 07 2012

Status

approved