%I #63 Aug 13 2023 08:48:27
%S 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,
%T 16,24,8,16,12,12,6,16,192,16,12,12,24,16,10,22,192,12,8,32,16,24,6,
%U 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
%N The order of the group Aut(Z/nZ)*, or the number of automorphisms of (Z/nZ)*.
%C (Z/nZ)* represents the multiplicative group of units mod n and this sequence gives the number of automorphisms of (Z/nZ)*.
%C A formula for this sequence can be found in the Hillar and Rhea reference.
%C 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
%H Andrew Howroyd, <a href="/A258615/b258615.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Dominic Milioto)
%H C. J. Hillar and D. Rhea, <a href="http://www.msri.org/people/members/chillar/files/autabeliangrps.pdf">Automorphisms of finite abelian groups</a>
%H C. J. Hillar and D. Rhea, <a href="http://arxiv.org/abs/math/0605185">Automorphisms of finite abelian groups</a>, arXiv:math/0605185 [math.GR], 2006.
%H Dominic Milioto, <a href="http://demonstrations.wolfram.com/TheSizeOfAnAutomorphismGroup/">The size of an Automorphism Group</a>, Wolfram Demonstrations Project.
%H Jianing Song, <a href="/A258615/a258615_1.txt">Structure and SmallGroupId of Aut((Z/nZ)*) for n <= 200</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>
%F See Theorem 4.1 in the Hillar and Rhea link.
%F a(p^k) = A000010(A000010(p^k)) for p an odd prime and k>0. - _Tom Edgar_, Jun 05 2015
%e |Aut((Z/1Z)*)|=1. |Aut(Z/28Z)*| = 12.
%e (Z/5Z)* is isomorphic to Z/4Z, which has two automorphisms, so a(5) = 2. - _Tom Edgar_, Jun 05 2015
%o (PARI)
%o zp(g)={sum(i=1, #g, my(f=factor(g[i])); sum(j=1, #f~, x^f[j,1]*y^f[j,2]))}
%o 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 + (i-1)*c*(d+c); m*=prod(i=1, c, p^i-1); d+=c)); s+=d*(d-1)/2; m*p^s}
%o a(n)={my(p=zp(znstar(n).cyc)); prod(i=1, poldegree(p), aut(i, polcoeff(p, i)))} \\ _Andrew Howroyd_, Jun 30 2018
%Y Cf. A000010.
%K nonn
%O 1,5
%A _Dominic Milioto_, Jun 05 2015