

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
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OFFSET

1,3


COMMENTS

Multiplicative with a(2^p) = 1, a(p^e) = (e+1)*p^e + (2*(1+(e*pe1)*p^e))/((p1)^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 12581276.
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), 13001306 (Abstract, pdf, ps).
Index entries for sequences related to D_4 lattice
Index entries for sequences related to sublattices


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))/(p1)^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



