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%I #30 Apr 23 2019 02:00:22
%S 1,1,8,1,12,8,16,1,41,12,24,8,28,16,96,1,36,41,40,12,128,24,48,8,97,
%T 28,176,16,60,96,64,1,192,36,192,41,76,40,224,12,84,128,88,24,492,48,
%U 96,8,177,97,288,28,108,176,288,16,320,60,120,96,124,64,656,1
%N Number of similar sublattices of index n^2 in root lattice D_4.
%C Multiplicative with a(2^p) = 1, a(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 Michael De Vlieger, <a href="/A045771/b045771.txt">Table of n, a(n) for n = 1..16384</a>
%H M. Baake and R. 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 Michael Baake and Peter Zeiner, "Similar Sublattices", Ch. 3.5 in <a href="https://doi.org/10.1017/9781139033862">Aperiodic Order</a>, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105.
%H J. 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/Da#D4">Index entries for sequences related to D_4 lattice</a>
%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%t Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[1 <= p <= 2, 1, (e + 1) p^e + (2 (1 + (e p - e - 1)*p^e))/((p - 1)^2)]] &, 64] (* _Michael De Vlieger_, Mar 02 2018 *)
%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)); prod(i=1, #f~, fp(f[i, 1], f[i, 2]));} \\ _Michel Marcus_, Mar 03 2014
%Y Cf. A035292.
%K nonn,mult
%O 1,3
%A Michael Baake (baake(AT)miles.math.ualberta.ca)
%E More terms from _Michel Marcus_, Mar 03 2014
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