cp's OEIS Frontend

A064767 Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

%I A064767 #52 Mar 04 2025 21:54:52
%S A064767 1,168,11232,86016,1488000,1886976,33784128,44040192,221079456,
%T A064767 249984000,2124276000,966131712,9726417792,5675733504,16713216000,
%U A064767 22548578304,111203278848,37141348608,304812862560,127991808000
%N A064767 Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).
%C A064767 Also number of 3 X 3 invertible matrices over the ring Z/nZ. - _Max Alekseyev_, Nov 02 2007
%C A064767 Order of the group GL(3,Z_n). For n > 2, a(n) is divisible by 96. - _Jianing Song_, Nov 24 2018
%H A064767 T. D. Noe, <a href="/A064767/b064767.txt">Table of n, a(n) for n = 1..1000</a>
%H A064767 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A064767 C. J. Hillar and D. L. Rhea, <a href="http://arxiv.org/abs/math/0605185">Automorphisms of finite abelian groups</a>, arXiv:math/0605185 [math.GR], 2006.
%H A064767 C. J. Hillar and D. L. Rhea, <a href="http://www.jstor.org/stable/27642365">Automorphisms of finite abelian groups</a>, Amer. Math. Monthly 114 (2007), no 10, 917-923.
%H A064767 J. Overbey, W. Traves and J. Wojdylo, <a href="http://jeff.over.bz/papers/undergrad/on-the-keyspace-of-the-hill-cipher.pdf">On the Keyspace of the Hill Cipher</a>, Cryptologia, Vol. 29 , Iss. 1, 2005.
%H A064767 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>.
%F A064767 a(n) = phi(n)*A011785(n). - _Vladeta Jovovic_, Oct 29 2001
%F A064767 a(n) = n^9*Product_{primes p dividing n} ((1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). This also gives a formula for A011785.
%F A064767 Multiplicative with a(p^e) = p^(9*e-6)*(p^3 - 1)*(p^2 - 1)*(p - 1). - _Vladeta Jovovic_, Nov 18 2001
%F A064767 Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} ((p^7 - p^5 - p^4 + p^2 + p - 1)/p^7) =  0.05123382571... . - _Amiram Eldar_, Oct 23 2022
%t A064767 a[n_] := n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]]; a[1] = 1; Array[a, 20] (* _Jean-François Alcover_, Mar 21 2017 *)
%o A064767 (PARI) a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k))); \\ _Michel Marcus_, Jun 30 2015
%o A064767 (PARI) a(n,f=factor(n))=prod(i=1,#f~, ((1 - 1/f[i,1]^3)*(1 - 1/f[i,1]^2)*(1 - 1/f[i,1])))*n^9 \\ _Charles R Greathouse IV_, Mar 04 2025
%o A064767 (Python)
%o A064767 from math import prod
%o A064767 from sympy import factorint
%o A064767 def A064767(n): return prod(p**(3*(3*e-2))*(p*(p*(p**2*(p*(p-1)-1)+1)+1)-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Mar 04 2025
%Y A064767 Row n=3 of A316622.
%Y A064767 Cf. A000010, A064803.
%Y A064767 Cf. A000252 (GL(2,Z_n)), A305186 (GL(4,Z_n)).
%Y A064767 Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
%K A064767 nonn,easy,mult
%O A064767 1,2
%A A064767 Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 24 2001
%E A064767 More terms from _Vladeta Jovovic_, Nov 18 2001