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
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 and 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
Crossrefs
Programs
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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 *)
-
PARI
a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z ) ) ); /* Joerg Arndt, Oct 07 2012 */
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).
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
Conjectured g.f.: Sum_{k>=1} Sum {l>=1} Sum {m>=1} k*l^2*m^3*x^(k*l*m)/(1 - x^(k*l*m)) (by extension of g.f for A001001). - Miles Wilson, Apr 05 2025
Extensions
Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011
More terms from Joerg Arndt, Oct 07 2012