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).

2, 2187, 15625, 823543, 1771561, 62748517, 24137569, 893871739, 148035889, 594823321, 27512614111, 94931877133, 4750104241, 271818611107, 10779215329, 22164361129, 42180533641, 3142742836021, 6060711605323, 128100283921, 11047398519097, 19203908986159, 326940373369

Offset: 1

Author

Jianing Song, Aug 25 2023

Keywords

• nonn
• easy

Comments

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).

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).

Examples

			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.
It seems that |Aut(G)| = 5^r => |Aut(G)| >= 15625 = 3^6, and |Aut(G)| = 15625 => G = SmallGroup(3125,38).
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.
It seems that |Aut(G)| = 11^r => |Aut(G)| >= 1771561 = 11^6, and |Aut(G)| = 1771561 => G = SmallGroup(161051,40).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Peter Hegarty and Desmond MacHale, Minimal odd order automorphism groups, arXiv:0905.0993 [math.GR], 2009.

Crossrefs

Cf. A030516 (sixth powers of primes), A092759 (seventh powers of primes).

Programs

PARI

a(n) = if(n==1, 2, my(p=prime(n)); if(p%3==2, p^6, p^7))

Python

from sympy import prime
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