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 A045771 Number of similar sublattices of index n^2 in root lattice D_4. 2
 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, 28, 176, 16, 60, 96, 64, 1, 192, 36, 192, 41, 76, 40, 224, 12, 84, 128, 88, 24, 492, 48, 96, 8, 177, 97, 288, 28, 108, 176, 288, 16, 320, 60, 120, 96, 124, 64, 656, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..16384 M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. (1999), 51 1258-1276. Michael Baake and Peter Zeiner, "Similar Sublattices", Ch. 3.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105. 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). MATHEMATICA 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 *) 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)); prod(i=1, #f~, fp(f[i, 1], f[i, 2])); } \\ Michel Marcus, Mar 03 2014 CROSSREFS Cf. A035292. Sequence in context: A032012 A092702 A070475 * A070488 A322079 A124906 Adjacent sequences:  A045768 A045769 A045770 * A045772 A045773 A045774 KEYWORD nonn,mult AUTHOR Michael Baake (baake(AT)miles.math.ualberta.ca) EXTENSIONS More terms from Michel Marcus, Mar 03 2014 STATUS approved

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Last modified November 28 14:27 EST 2020. Contains 338724 sequences. (Running on oeis4.)