OFFSET
1,1
COMMENTS
By non-prime-power, we mean a member of A024619: a number not of the form p^m, m >= 0.
The condition is easily satisfied for prime power orders for every cyclic group C of order n. Since |Aut(C)| = phi(n), if n = p^m then |Aut(C)| = (p-1)*p^(m-1), and GCD(|Aut(C)|,p^m) = p^(m-1). Hence |Aut(C)|/GCD(|Aut(G)|,p^m) = (p-1)*p^(m-1)/p^(m-1) = p-1, and p^m/GCD(|Aut(G)|,p^m) = p^m/p^(m-1) = p.
Are all terms even?
EXAMPLE
18 is a term since |Aut(C3xS3)|/GCD(|Aut(C3xS3)|,18) = 2 and 18/GCD(|Aut(C3xS3)|,18) = 3, and 18 is not a prime power.
PROG
(GAP)
S:=[];;
for n in [1..200] do
for i in [1..NrSmallGroups(n)] do
Aut:=Order(AutomorphismGroup(SmallGroup(n, i)));;
if Length(Unique(Factors(n)))=1 then
break;
fi;
if Aut/GCD_INT(n, Aut)=(n/GCD_INT(n, Aut))-1 then
S:=Concatenation(S, [n]);
break;
fi;
od;
od;
Print(S);
CROSSREFS
KEYWORD
nonn
AUTHOR
Miles Englezou, Aug 04 2024
STATUS
approved