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A374666
Orders of groups for which there exists at least one group G of order k such that |Aut(G)|/GCD(|Aut(G)|,k) = (k/GCD(|Aut(G)|,k))-1.
1
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 24, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 54, 56, 59, 60, 61, 64, 66, 67, 71, 72, 73, 78, 79, 80, 81, 83, 84, 88, 89, 96, 97, 100, 101, 102, 103, 104, 107, 108, 109, 112, 113, 114, 120
OFFSET
1,1
COMMENTS
The single group of prime order p has an automorphism group of order p-1. a(n) is a generalization of this.
EXAMPLE
3 is a term since |Aut(C3)|/GCD(|Aut(C3)|,3) = 2 and 3/GCD(|Aut(C3)|,3) = 3.
40 is a term since |Aut(C5xD8)|/GCD(|Aut(C5xD8)|,40) = 4 and 40/GCD(|Aut(C5xD8)|,40) = 5.
PROG
(GAP)
S:=[];;
for n in [1..30] do
for i in [1..NrSmallGroups(n)] do
Aut:=Order(AutomorphismGroup(SmallGroup(n, i)));;
if Aut/GCD_INT(n, Aut)=(n/GCD_INT(n, Aut))-1 then
S:=Concatenation(S, [n]);
break;
fi;
od;
od;
Print(S);
CROSSREFS
Cf. A246655 (a subsequence).
Sequence in context: A062491 A087092 A046684 * A082377 A166401 A329131
KEYWORD
nonn
AUTHOR
Miles Englezou, Jul 15 2024
STATUS
approved