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%I #48 May 08 2023 21:50:22
%S 1,15,40,155,156,600,400,1395,1210,2340,1464,6200,2380,6000,6240,
%T 11811,5220,18150,7240,24180,16000,21960,12720,55800,20306,35700,
%U 33880,62000,25260,93600,30784,97155,58560,78300,62400,187550,52060,108600,95200,217620,70644,240000,81400
%N Number of sublattices of index n in generic 4-dimensional lattice.
%D M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
%H Amiram Eldar, <a href="/A038991/b038991.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from G. C. Greubel)
%H M. Baake and U. Grimm, <a href="http://arXiv.org/abs/math-ph/0212015">Combinatorial problems of (quasi)crystallography</a>, arXiv:math-ph/0212015, 2002.
%H M. Baake, N. Neumarker, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Baake/baake7.html">A Note on the Relation Between Fixed Point and Orbit Count Sequences</a>, JIS 12 (2009) 09.4.4, Section 3.
%H Tad White, <a href="http://arxiv.org/abs/1304.2830">Counting Free Abelian Actions</a>, arXiv preprint arXiv:1304.2830 [math.CO], 2013.
%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%F f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=4.
%F Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3).
%F Dirichlet convolution of A000578 and A001001.
%F Multiplicative with a(p^e) = Product_{k=1..3} (p^(e+k)-1)/(p^k-1).
%F Sum_{k=1..n} a(k) ~ Pi^6 * Zeta(3) * n^4 / 2160. - _Vaclav Kotesovec_, Feb 01 2019
%t a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #&]&]&]; Array[a, 50] (* _Jean-François Alcover_, Dec 02 2015, after _Joerg Arndt_ *)
%t 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 *)
%o (PARI) a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z ) ) ); /* _Joerg Arndt_, Oct 07 2012 */
%Y Column 4 of A160870.
%Y Cf. A001001, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_
%E Offset changed from 0 to 1 by _R. J. Mathar_, Mar 31 2011
%E More terms from _Joerg Arndt_, Oct 07 2012
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