

A035292


Number of similar sublattices of Z^4 of index n^2.


1



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

1,2


COMMENTS

Multiplicative with a(2^p) = 3, a(p^e) = (e+1)*p^e + (e+1)*p^e + (2*(1+(e*pe1)*p^e))/((p1)^2), p > 2.  Christian G. Bower, May 21 2005


LINKS

Table of n, a(n) for n=1..62.
Research Group Michael Baake, Preprints & Recent Articles: Algebra, Combinatorics and Number Theory
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG].
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. (1999), 51 12581276.
John H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 13001306 (Abstract, pdf, ps).
Index entries for sequences related to sublattices


FORMULA

Baake and Moody give Dirichlet generating function.
For odd n, a(n) = A045771(n); for even n, a(n) = 3*A045771(n).  Michel Marcus, Mar 03 2014


MATHEMATICA

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 *)


PROG

(PARI) fp(p, e) = if (p % 2, (e+1)*p^e + 2*(1(e+1)*p^e+e*p^(e+1))/(p1)^2, 1);
a(n) = {my(f = factor(n)); a045771 = prod(i=1, #f~, fp(f[i, 1], f[i, 2])); if (n % 2, a045771, 3*a045771); } \\ Michel Marcus, Mar 03 2014


CROSSREFS

Cf. A045771.
Sequence in context: A016623 A046543 A233129 * A144457 A220138 A146975
Adjacent sequences: A035289 A035290 A035291 * A035293 A035294 A035295


KEYWORD

nonn,mult


AUTHOR

Michael Baake (michael.baake(AT)unituebingen.de)


STATUS

approved



