A045771 Number of similar sublattices of index n^2 in root lattice D_4.

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

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).

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))/(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 A349142 A322079

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