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A138565 Array read by rows: T(n,k) is the number of automorphisms of the k-th Abelian group of order n, where the ordering is such that the rows are nondecreasing. 0
1, 1, 2, 2, 6, 4, 2, 6, 4, 8, 168, 6, 48, 4, 10, 4, 12, 12, 6, 8, 8, 16, 96, 192, 20160, 16, 6, 48, 18, 8, 24, 12, 10, 22, 8, 16, 336, 20, 480, 12, 18, 108, 11232, 12, 36, 28, 8, 30, 16, 32, 128, 384, 1536, 21504, 9999360, 20, 16, 24, 12, 36, 96, 288, 36, 18, 24, 16, 32, 672 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is a subtable of A137316.

The length of the n-th row is A000688(n).

The largest value of the n-th row is A061350(n).

The number phi(n) = A000010(n) appears in the n-th row.

The number A064767(n) appears in the (n^3)-th row.

The number A062771(n) appears in the (2n)-th row.

LINKS

Table of n, a(n) for n=1..68.

C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.

C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

EXAMPLE

The table begins as follows:

1

1

2

2 6

4

2

6

4 8 168

6 48

4

10

4 12

The first row with two numbers corresponds to the two Abelian groups of order 4, the cyclic group C_4 and the Klein group C_2 x C_2, whose automorphism groups are respectively the group (C_4)^x = C_2 and the symmetric group S_3.

PROG

(GAP4)

Print("\n") ;

for o in [ 1 .. 40 ] do

    n := NumberSmallGroups(o) ;

    og := [] ;

    for i in [1 .. n] do

        g := SmallGroup(o, i) ;

        if IsAbelian(g) then

            H := AutomorphismGroup(g) ;

            ho := Order(H) ;

            Add(og, ho) ;

        fi ;

    od;

    Sort(og) ;

    Print(og) ;

    Print("\n") ;

od; # R. J. Mathar, Jul 13 2013

CROSSREFS

Sequence in context: A324349 A092384 A061915 * A137316 A064851 A305353

Adjacent sequences:  A138562 A138563 A138564 * A138566 A138567 A138568

KEYWORD

easy,nonn,tabf

AUTHOR

Benoit Jubin, May 12 2008

STATUS

approved

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Last modified May 31 15:41 EDT 2020. Contains 334748 sequences. (Running on oeis4.)