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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

T. D. Noe, Table of n, a(n) for n=1..1024

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-Fran├žois Alcover, May 21 2012, after Maple *)

CROSSREFS

Cf. A059773, A002884, A053290, A053292, A053293.

Sequence in context: A021795 A008904 A074382 * A046276 A003571 A068457

Adjacent sequences:  A061347 A061348 A061349 * A061351 A061352 A061353

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 24 04:15 EDT 2017. Contains 283984 sequences.