A038991 Number of sublattices of index n in generic 4-dimensional lattice.

1, 15, 40, 155, 156, 600, 400, 1395, 1210, 2340, 1464, 6200, 2380, 6000, 6240, 11811, 5220, 18150, 7240, 24180, 16000, 21960, 12720, 55800, 20306, 35700, 33880, 62000, 25260, 93600, 30784, 97155, 58560, 78300, 62400, 187550, 52060, 108600, 95200, 217620, 70644, 240000, 81400

Offset

1,2

References

M. Baake, "Solution of coincidence problem...", 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 U. Grimm, Combinatorial problems of (quasi)crystallography, arXiv:math-ph/0212015, 2002.

M. Baake, N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.

Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

Index entries for sequences related to sublattices

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=4.

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3).

Dirichlet convolution of A000578 and A001001.

Multiplicative with a(p^e) = Product_{k=1..3} (p^(e+k)-1)/(p^k-1).

Sum_{k=1..n} a(k) ~ Pi^6 * Zeta(3) * n^4 / 2160. - Vaclav Kotesovec, Feb 01 2019

Mathematica

a[n_] := DivisorSum[n, #*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, 3}]; 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 ) ) ); /* Joerg Arndt, Oct 07 2012 */

Crossrefs

Column 4 of A160870.

Cf. A001001, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

Sequence in context: A160891 A223425 A175926 * A068020 A131991 A116042

Adjacent sequences: A038988 A038989 A038990 * A038992 A038993 A038994

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