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A finite group G is called complete if Aut G = Inn G and Z(G) = {1} i.e. G has no outer automorphisms and the center of G is trivial.

The symmetric group S(n) of order n! is complete for n not equal to 2 or 6.

If p is an odd prime, there is a complete group of order p(p-1) and a complete group of order p^m*(p^m - p^(m-1)) for each m.

Dark in 1975 discovered a nontrivial complete group G of odd order. It has order 788953370457 = 3*19*7^12. [Corrected by Jianing Song, Aug 25 2023]

Recently, Dark showed that the smallest possible nontrivial complete group G of odd order has order 352947 = 3*7^6. [In fact, for every prime p == 1 (mod 3), there exists a complete group of order 3*p^6, and it occurs as the automorphism group of a group of order 3*p^5. This means that there are infinitely many odd terms in this sequence. See the M. John Curran and Rex S. Dark link. - Jianing Song, Aug 25 2023]

From Jianing Song, Aug 25 2023: (Start)

The holomorph (see the Wikipedia link) of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link.

No prime power (A246655) is a term. See the first Groupprops link.

The automorphism group of a complete group is isomorphic to itself. The converse is not true, as shown by the counterexamples D_8 and D_12. In contrast with the fact that the holomorph of a complete group is isomorphic to the external direct product of two copies of it (see the second Groupprops link), the holomorph of D_8 (SmallGroup(64,134)) is not isomorphic to D_8 X D_8 = SmallGroup(64,226), and the holomorph of D_12 (SmallGroup(144,154)) is not isomorphic to D_12 X D_12 = SmallGroup(144,192). (End)

By Lagrange's Theorem a(n) is a multiple of n.

a(n) is 6n if n is odd and 4n if n is even. This is because the groups involved are C(n) X S3 if n is odd, where S3 is the symmetric group of order 6, and C(n/2) X D8 if n is even, where D8 is the dihedral group of order 8 and C(m) is the cyclic group of order m.

By Lagrange's Theorem a(n) is a multiple of n.

This is the number of choices for the group G defined in A340512.

By Lagrange's theorem, a(n) is always a multiple of n, and it is likely this multiple is always 2, 3, or 4 for n>1.

Because of dihedral groups, a(2k+1) = 4k+2.

Every multiple of a(n) is also a term of the sequence because the direct product of a group G with any Abelian group A satisfies (GXA)' = G'.

Since a(1)= 16 and p^3 is in the sequence for any odd prime p, by taking direct products with cyclic groups we see that n belongs to the sequence if either 16 or p^3 divides n for an odd prime p. However, 72 and 147, which are not of this form, both belong to the sequence. Also, every multiple of each term in the sequence is also a term of the sequence. Conformality of groups is an equivalence relation but there seem to be virtually no known conformality invariants other than group order.