

A258615


The order of the group Aut(Z/nZ)*, or the number of automorphisms of (Z/nZ)*.


4



1, 1, 1, 1, 2, 1, 2, 6, 2, 2, 4, 6, 4, 2, 8, 8, 8, 2, 6, 8, 12, 4, 10, 168, 8, 4, 6, 12, 12, 8, 8, 16, 24, 8, 16, 12, 12, 6, 16, 192, 16, 12, 12, 24, 16, 10, 22, 192, 12, 8, 32, 16, 24, 6, 32, 336, 36, 12, 28, 192, 16, 8, 288, 32, 192, 24, 20, 32, 60, 16, 24, 336, 24, 12, 32, 36, 48, 16, 24, 1536, 18, 16, 40, 336, 256
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OFFSET

1,5


COMMENTS

(Z/nZ)* represents the multiplicative group of units mod n and this sequence gives the number of automorphisms of (Z/nZ)*.
A formula for this sequence can be found in the Hillar and Rhea reference.
Or equivalently, a(n) is the order of Aut(Aut(C_n)), where C_n is the cyclic group of order n.  Jianing Song, Apr 06 2019


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000 (terms 1..500 from Dominic Milioto)
C. J. Hillar and D. Rhea, Automorphisms of finite abelian groups
C. J. Hillar and D. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
Dominic Milioto, The size of an Automorphism Group, Wolfram Demonstrations Project.
Wikipedia, Multiplicative group of integers modulo n


FORMULA

See Theorem 4.1 in the Hillar and Rhea link.
a(p^k) = A000010(A000010(p^k)) for p an odd prime and k>0.  Tom Edgar, Jun 05 2015


EXAMPLE

Aut((Z/1Z)*)=1. Aut(Z/28Z)* = 12.
(Z/5Z)* is isomorphic to Z/4Z, which has two automorphisms, so a(5) = 2.  Tom Edgar, Jun 05 2015


PROG

(PARI)
zp(g)={sum(i=1, #g, my(f=factor(g[i])); sum(j=1, #f~, x^f[j, 1]*y^f[j, 2]))}
aut(p, q)={my(s=0, d=0, m=1); forstep(i=poldegree(q), 1, 1, my(c=polcoeff(q, i)); if(c, s+=i*c*d + (i1)*c*(d+c); m*=prod(i=1, c, p^i1); d+=c)); s+=d*(d1)/2; m*p^s}
a(n)={my(p=zp(znstar(n).cyc)); prod(i=1, poldegree(p), aut(i, polcoeff(p, i)))} \\ Andrew Howroyd, Jun 30 2018


CROSSREFS

Cf. A000010.
Sequence in context: A265870 A341409 A284466 * A152431 A143965 A182073
Adjacent sequences: A258612 A258613 A258614 * A258616 A258617 A258618


KEYWORD

nonn


AUTHOR

Dominic Milioto, Jun 05 2015


STATUS

approved



