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Let u_n be the group of units mod n (invertible elements in the ring Z_n); a(n) is the number of cyclic subgroups in u_n.
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%I #8 Jul 01 2018 21:18:44

%S 1,1,2,2,3,2,4,4,4,3,4,4,6,4,6,6,5,4,6,6,8,4,4,8,6,6,6,8,6,6,8,8,8,5,

%T 12,8,9,6,12,12,8,8,8,8,12,4,4,12,8,6,10,12,6,6,12,16,12,6,4,12,12,8,

%U 20,10,20,8,8,10,8,12,8,16,12,9,12,12,16,12,8,20

%N Let u_n be the group of units mod n (invertible elements in the ring Z_n); a(n) is the number of cyclic subgroups in u_n.

%H Andrew Howroyd, <a href="/A066589/b066589.txt">Table of n, a(n) for n = 1..1000</a>

%o (GAP) List([1..80], n->Sum( ConjugacyClasses( AutomorphismGroup( CyclicGroup(n))), x->Size(x)/Phi(Order( Representative(x)))));

%o (PARI) a(n)={sum(i=1, n, if(gcd(i,n)==1, 1/eulerphi(znorder(Mod(i,n)))))} \\ _Andrew Howroyd_, Jul 01 2018

%K nonn

%O 1,3

%A Sharon Sela (sharonsela(AT)hotmail.com), Jan 08 2002

%E a(32)-a(33) inserted and terms a(37) and beyond from _Andrew Howroyd_, Jul 01 2018