login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A035292 Number of similar sublattices of Z^4 of index n^2. 1
1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, 24, 28, 48, 96, 3, 36, 123, 40, 36, 128, 72, 48, 24, 97, 84, 176, 48, 60, 288, 64, 3, 192, 108, 192, 123, 76, 120, 224, 36, 84, 384, 88, 72, 492, 144, 96, 24, 177, 291, 288, 84, 108, 528, 288, 48, 320, 180, 120, 288, 124, 192 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Multiplicative with a(2^p) = 3, a(p^e) = (e+1)*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

Table of n, a(n) for n=1..62.

Research Group Michael Baake, Preprints & Recent Articles: Algebra, Combinatorics and Number Theory

Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG].

Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. (1999), 51 1258-1276.

John 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 sublattices

FORMULA

Baake and Moody give Dirichlet generating function.

For odd n, a(n) = A045771(n); for even n, a(n) = 3*A045771(n). - Michel Marcus, Mar 03 2014

MATHEMATICA

Clear[ a, f ]; a[ {p_, r_} ] := If[ p == 2, 3, (r + 1)*p^r + (2*(1 - (r + 1)*p^r + r*p^(r + 1)))/(p - 1)^2 ]; f[ m_Integer ] := f[ m ] = Times @@ a /@ FactorInteger[ m ]; (* f[ m ] is number of similar sublattices of Z^4 of index m^2 *)

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)); a045771 = prod(i=1, #f~, fp(f[i, 1], f[i, 2])); if (n % 2, a045771, 3*a045771); } \\ Michel Marcus, Mar 03 2014

CROSSREFS

Cf. A045771.

Sequence in context: A273100 A046543 A233129 * A144457 A220138 A146975

Adjacent sequences:  A035289 A035290 A035291 * A035293 A035294 A035295

KEYWORD

nonn,mult

AUTHOR

Michael Baake (michael.baake(AT)uni-tuebingen.de)

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 4 17:40 EST 2016. Contains 278755 sequences.