A137315 -id:A137315 - cp's OEIS frontend

A340521 List of possible orders of automorphism groups of finite groups.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60

Offset: 1

Des MacHale, Feb 05 2021

The terms shown here match the initial terms of all of A002174, A002202, A049225, but this is a strictly different sequence since it is known that it contains 3^7 = 2187 (which is the smallest odd term greater than 1), whereas for the other three sequences all terms greater than 1 are even.
This is a supersequence of A002202 since |Aut(Z/nZ)| = phi(n). - Jianing Song, Feb 05 2021
John Bray has produced a group G of order 3^2*19 = 171 such that |Aut G| = 1026 = 2.3^3.19. So 1026 is in the present sequence but is not in A002202. So the present sequence contains both odd and even terms not in A002202. - Des MacHale, Feb 10 2021
For more about this problem, see the references in A137315.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

Cf. A002174, A002202, A049225, A137315.

A139795 Least m such that k>=m implies phi(k)>=n (where phi is the Euler totient function, sequence A000010).

Original entry on oeis.org

1, 3, 7, 7, 13, 13, 19, 19, 31, 31, 31, 31, 43, 43, 43, 43, 61, 61, 61, 61, 67, 67, 67, 67, 91, 91, 91, 91, 91, 91, 91, 91, 121, 121, 121, 121, 127, 127, 127, 127, 151, 151, 151, 151, 151, 151, 151, 151, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211

Offset: 1

Benoit Jubin, May 21 2008

nonn

Define b(n)=A006511(m)+1 where m is the unique integer such that A002202(m)A002202(m+1) (with the convention A002202(0)=A006511(0)=0). Then a(1)=b(1) and a(n+1)=max(a(n),b(n+1)).

The sequence a(n) without the repetitions is 1+A036913(n).

			a(5)=13 because if k>=13, then phi(k)>=5, but phi(12)=4.

Max Alekseyev, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Different from A137315 (see Comments in that entry).

{m=0;for(n=1,100,print1(m+1,",");trap(,0,m=max(m,vecmax(invphi(n)))))}

A341825 Number of finite groups G with |Aut(G)| = n.

Original entry on oeis.org

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

Offset: 1

Des MacHale, Mar 02 2021

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.

			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.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

Cf. A002202, A137315, A340521, A341823, A341824.

a(2) = 3, a(p) = 0 if p odd prime.

a(A002202(n)) > 0, since |Aut(C_n)| = phi(n).

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A340521 List of possible orders of automorphism groups of finite groups.

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A139795 Least m such that k>=m implies phi(k)>=n (where phi is the Euler totient function, sequence A000010).

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A341825 Number of finite groups G with |Aut(G)| = n.

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