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A038995
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Number of sublattices of index n in generic 8-dimensional lattice.
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12
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1, 255, 3280, 43435, 97656, 836400, 960800, 6347715, 8069620, 24902280, 21435888, 142466800, 67977560, 245004000, 320311680, 866251507, 435984840, 2057753100, 943531280, 4241688360, 3151424000, 5466151440, 3559590240, 20820505200, 7947261556, 17334277800, 18326727760
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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FORMULA
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f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=8.
Multiplicative with a(p^e) = Product_{k=1..7} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)/43401015000 = 0.285716... . - Amiram Eldar, Oct 19 2022
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MATHEMATICA
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f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 7}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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