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A035292
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Number of similar sublattices of Z^4 of index n^2.
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1
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1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, 24, 28, 48, 96, 3, 36, 123, 40, 36, 128, 72, 48, 24, 97, 84, 176, 48, 60, 288, 64, 3, 192, 108, 192, 123, 76, 120, 224, 36, 84, 384, 88, 72, 492, 144, 96, 24, 177, 291, 288, 84, 108, 528, 288, 48, 320, 180, 120, 288, 124, 192
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OFFSET
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1,2
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COMMENTS
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Multiplicative with a(2^p) = 3, a(p^e) = (e+1)*p^e + (e+1)*p^e + (2*(1+(e*p-e-1)*p^e))/((p-1)^2), p>2. Christian G. Bower May 21, 2005.
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LINKS
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Table of n, a(n) for n=1..62.
M. Baake, Algebra, Combinatorics and Number Theory
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Index entries for sequences related to sublattices
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FORMULA
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Baake and Moody give Dirichlet generating function.
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MATHEMATICA
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Clear[ a, f ]; a[ {p_, r_} ] := If[ p == 2, 3, (r + 1)*p^r + (2*(1 - (r + 1)*p^r + r*p^(r + 1)))/(p - 1)^2 ]; f[ m_Integer ] := f[ m ] = Times @@ a /@ FactorInteger[ m ]; (* f[ m ] is number of similar sublattices of Z^4 of index m^2 *)
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CROSSREFS
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Cf. A045771.
Sequence in context: A205126 A016623 A046543 * A144457 A220138 A146975
Adjacent sequences: A035289 A035290 A035291 * A035293 A035294 A035295
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Baake (michael.baake(AT)uni-tuebingen.de)
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STATUS
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approved
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