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A365179 a(1) = 2; for n >= 2, a(n) = p^6 if p == 2 (mod 3), p^7 if p = 3 or p == 1 (mod 3), where p = prime(n).

%I A365179 #25 Aug 27 2023 04:21:18
%S A365179 2,2187,15625,823543,1771561,62748517,24137569,893871739,148035889,
%T A365179 594823321,27512614111,94931877133,4750104241,271818611107,
%U A365179 10779215329,22164361129,42180533641,3142742836021,6060711605323,128100283921,11047398519097,19203908986159,326940373369
%N A365179 a(1) = 2; for n >= 2, a(n) = p^6 if p == 2 (mod 3), p^7 if p = 3 or p == 1 (mod 3), where p = prime(n).
%C A365179 Conjecture 1: a(n) is the smallest nontrivial power of p such that there exists a finite nontrivial group whose automorphism group is of order a(n).
%C A365179 Conjecture 2: for n >= 2, if |Aut(G)| = a(n), then |G| = a(n)/p, where p = prime(n). Moreover, G is unique up to isomorphism if p == 2 (mod 3).
%H A365179 Jianing Song, <a href="/A365179/b365179.txt">Table of n, a(n) for n = 1..10000</a>
%H A365179 Peter Hegarty and Desmond MacHale, <a href="https://arxiv.org/abs/0905.0993">Minimal odd order automorphism groups</a>, arXiv:0905.0993 [math.GR], 2009.
%e A365179 By the Peter Hegarty and Desmond MacHale link we have |Aut(G)| = 3^r => |Aut(G)| = 2187 = 3^7. It seems that if |Aut(G)| = 2187, then G = SmallGroup(729,m) for m = 90, 92 or 414.
%e A365179 It seems that |Aut(G)| = 5^r => |Aut(G)| >= 15625 = 3^6, and |Aut(G)| = 15625 => G = SmallGroup(3125,38).
%e A365179 It seems that |Aut(G)| = 7^r => |Aut(G)| >= 823543 = 7^7, and |Aut(G)| = 823543 => G = SmallGroup(117649,m) for m = 199, 824, 831 through 836.
%e A365179 It seems that |Aut(G)| = 11^r => |Aut(G)| >= 1771561 = 11^6, and |Aut(G)| = 1771561 => G = SmallGroup(161051,40).
%o A365179 (PARI) a(n) = if(n==1, 2, my(p=prime(n)); if(p%3==2, p^6, p^7))
%o A365179 (Python)
%o A365179 from sympy import prime
%o A365179 def A365179(n): return 2 if n == 1 else (p:=prime(n))**(6 if p%3 == 2 else 7) # _Chai Wah Wu_, Aug 26 2023
%Y A365179 Cf. A030516 (sixth powers of primes), A092759 (seventh powers of primes).
%K A365179 nonn,easy
%O A365179 1,1
%A A365179 _Jianing Song_, Aug 25 2023