A350586 Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 82, 86, 93, 94, 105, 106, 111, 118, 122, 129, 134, 142, 146, 155, 158, 165, 166, 178, 183, 194, 195, 201, 202, 203, 205, 206, 214, 218, 219, 226, 231, 237, 253, 254, 262, 274, 278, 285, 291, 298, 301, 302

Offset: 1

Author

Bernard Schott, Jan 07 2022

Keywords

• nonn

Comments

Differs from A064899 that is a subsequence: a(20) = 105 while A064899(20) = 106.

When m = 2*p, p odd prime, abelian group is C_{2*p} and nonabelian group is D_{2*p} ~ C_p : C_2.

When m = p*q, p

In both cases, C, D mean cyclic, dihedral groups of the stated order; the symbols ~ and : mean isomorphic to and semidirect product respectively.

A number m is a term iff m is squarefree and m has exactly one pair of prime factors (p, q) such that q == 1 (mod p). - David Radcliffe, Jul 30 2025

Examples

			There is only one group of order 1, 2, 3, 5 and the two groups of order 4 are abelian; hence 6 is the smallest term because the two groups of order 6 are the abelian and cyclic group C_6, while the nonabelian group is the symmetric group S_3 isomorphic to dihedral group D_6.
The smallest odd term is 21, the two corresponding groups are C_21 and semi-direct product C_7 : C_3.
The smallest term of the form p*q*r, p < q < r primes, is 105, the two corresponding groups are C_105 and semi-direct product C_35 : C_3.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• J.H. Conway, H. Dietrich, and E.A. O’Brien, Counting Groups: Gnus, Moas, and other Exotica, The Mathematical Intelligencer 30, 6-15 (2008), doi:10.1007/BF02985731.

Crossrefs

Equals A054395 \ A350322.

Subsequence of A060650 and of A005117.

Cf. A000001, A064899.

Programs

PARI

is(n,f=factor(n))=my(p=f[,1],s); if(#p && vecmax(f[,2])>1, return(0)); for(i=2,#p, for(j=1,i-1, if(p[i]%p[j]==1 && s++>1, return(0)))); s==1 \\ Charles R Greathouse IV, Jan 08 2022

PARI

list(lim)=my(v=List()); forsquarefree(n=6,lim\1, my(p=n[2][,1],s); for(i=2,#p, for(j=1,i-1, if(p[i]%p[j]==1 && s++>1, next(3)))); if(s==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2022

Python

from sympy import factorint
def is_ok(m):
    f = factorint(m)
    if any(e > 1 for e in f.values()): return False # m must be squarefree
    return sum(q % p == 1 for p in f for q in f) == 1 # David Radcliffe, Jul 30 2025

Extensions

More terms from Jinyuan Wang, Jan 08 2022

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