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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.

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).

Dirk Frettlöh, "Similar Sublattices", Ch. 3.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105.

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 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 April 20 14:27 EDT 2019. Contains 322310 sequences. (Running on oeis4.)