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%I #22 Mar 03 2014 03:49:50
%S 1,3,8,3,12,24,16,3,41,36,24,24,28,48,96,3,36,123,40,36,128,72,48,24,
%T 97,84,176,48,60,288,64,3,192,108,192,123,76,120,224,36,84,384,88,72,
%U 492,144,96,24,177,291,288,84,108,528,288,48,320,180,120,288,124,192
%N Number of similar sublattices of Z^4 of index n^2.
%C 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
%H Research Group Michael Baake, <a href="http://www.mathematik.uni-bielefeld.de/baake/preprints.html#algebra">Preprints & Recent Articles: Algebra, Combinatorics and Number Theory</a>
%H Michael Baake and Robert V. Moody, <a href="http://arXiv.org/abs/math.MG/9904028">Similarity submodules and root systems in four dimensions</a>, arXiv:math/9904028 [math.MG].
%H Michael Baake and Robert V. Moody, <a href="http://dx.doi.org/10.4153/CJM-1999-057-0">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math. (1999), 51 1258-1276.
%H John H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (<a href="http://neilsloane.com/doc/sim.txt">Abstract</a>, <a href="http://neilsloane.com/doc/sim.pdf">pdf</a>, <a href="http://neilsloane.com/doc/sim.ps">ps</a>).
%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%F Baake and Moody give Dirichlet generating function.
%F For odd n, a(n) = A045771(n); for even n, a(n) = 3*A045771(n). - _Michel Marcus_, Mar 03 2014
%t 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 *)
%o (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);
%o 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
%Y Cf. A045771.
%K nonn,mult
%O 1,2
%A Michael Baake (michael.baake(AT)uni-tuebingen.de)
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