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%I #40 May 08 2023 21:50:18
%S 1,31,121,651,781,3751,2801,11811,11011,24211,16105,78771,30941,86831,
%T 94501,200787,88741,341341,137561,508431,338921,499255,292561,1429131,
%U 508431,959171,925771,1823451,732541,2929531,954305,3309747,1948705,2750971,2187581,7168161,1926221
%N Number of sublattices of index n in generic 5-dimensional lattice.
%D 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.
%H Amiram Eldar, <a href="/A038992/b038992.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from G. C. Greubel)
%H M. Baake and 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 <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=5.
%F Multiplicative with a(p^e) = Product_{k=1..4} (p^(e+k)-1)/(p^k-1).
%F 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
%F 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
%t a[n_] := DivisorSum[n, #*DivisorSum[#, #*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, 4}]; 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 * sumdiv(z,w, w ) ) ) ); /* _Joerg Arndt_, Oct 07 2012 */
%Y Column 5 of A160870.
%Y Cf. A001001, A038991, 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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