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A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n. 4
1, 1, 2, 6, 4, 2, 6, 168, 48, 4, 10, 12, 12, 6, 8, 20160, 16, 48, 18, 24, 12, 10, 22, 336, 480, 12, 11232, 36, 28, 8, 30, 9999360, 20, 16, 24, 288, 36, 18, 24, 672, 40, 12, 42, 60, 192, 22, 46, 40320, 2016, 480, 32, 72, 52, 11232, 40, 1008, 36, 28, 58, 48, 60, 30, 288 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n) is multiplicative; if n = p^m is a prime power the maximal size of Aut(G) is attained by the elementary Abelian group G =(C_p)^m and then Aut(G) is GL(m,p) and a(n) = (p^m - 1)*(p^m - p)*...*(p^m - p^(m-1)). For general n the maximum will be for the direct product of the (C_p)^m over the prime powers dividing n and then the automorphism group is the direct product of the GL(m,p).
Equivalently, maximal size of Aut(G) where G is a nilpotent group of order n. - Eric M. Schmidt, Feb 27 2013
LINKS
MAPLE
A061350 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(mul(ifactors(n)[2][i][1]^ifactors(n)[2][i][2] - ifactors(n)[2][i][1]^(j - 1), j = 1..ifactors(n)[2][i][2])): od: RETURN(ans) end:
MATHEMATICA
a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ (Product[#[[1]]^#[[2]] - #[[1]]^k, {k, 0, #[[2]]-1}]& /@ FactorInteger[n]); Table[a[n], {n, 1, 63}] (* Jean-François Alcover, May 21 2012, after Maple *)
CROSSREFS
Sequence in context: A008904 A334397 A074382 * A046276 A368516 A283614
KEYWORD
nonn,mult,nice,easy
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001
EXTENSIONS
More terms from Vladeta Jovovic, Jun 12 2001
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)