A035292 Number of similar sublattices of Z^4 of index n^2.

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

Offset

1,2

Comments

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

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 1258-1276.

John 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

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))/(p-1)^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: A273100 A046543 A233129 * A144457 A220138 A146975

Adjacent sequences: A035289 A035290 A035291 * A035293 A035294 A035295

Keyword

nonn,mult

Author

Michael Baake (michael.baake(AT)uni-tuebingen.de)

Status

approved