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A341825
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Number of finite groups G with |Aut(G)| = n.
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2
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2, 3, 0, 4, 0, 6, 0, 7, 0, 2, 0, 9, 0, 0, 0, 11, 0, 4, 0, 7, 0, 2, 0, 22, 0, 0, 0, 2, 0, 2, 0, 19, 0, 0, 0, 12, 0, 0, 0, 14, 0, 7, 0, 3, 0, 2, 0
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OFFSET
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1,1
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COMMENTS
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The smallest odd index n > 1 for which a(n) > 0 is 2187 = 3^7 (see A340521).
There exist even indices n that are not values taken by totient function phi (A002202) for which a(n) > 0. For example, John Bray has produced a group G that is the semidirect product 19:9 of order 3^2*19 = 171 such that |Aut(G)| = 1026 = 2*3^3*19.
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LINKS
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FORMULA
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a(2) = 3, a(p) = 0 if p odd prime.
a(A002202(n)) > 0, since |Aut(C_n)| = phi(n).
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EXAMPLE
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a(6) = 6, because there are six groups G with |Aut(G)| = 6. Four cyclic groups: Aut(C_7) = Aut(C_9) = Aut(C_14) = Aut(C_18) ~~ C_6, and also Aut(C_2 x C_2) = Aut(S_3) ~~ S_3, where ~~ stands for “isomorphic to”. - Bernard Schott, Mar 02 2021
a(8) = 7, because there are seven groups G with |Aut(G)| = 8.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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