cp's OEIS Frontend

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

%I A350586 #42 Aug 01 2025 11:09:22
%S A350586 6,10,14,21,22,26,34,38,39,46,55,57,58,62,74,82,86,93,94,105,106,111,
%T A350586 118,122,129,134,142,146,155,158,165,166,178,183,194,195,201,202,203,
%U A350586 205,206,214,218,219,226,231,237,253,254,262,274,278,285,291,298,301,302
%N A350586 Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.
%C A350586 Differs from A064899 that is a subsequence: a(20) = 105 while A064899(20) = 106.
%C A350586 When m = 2*p, p odd prime, abelian group is C_{2*p} and nonabelian group is D_{2*p} ~ C_p : C_2.
%C A350586 When m = p*q, p<q odd primes with p dividing q-1, abelian group is C_{p*q} and nonabelian group is C_q : C_p.
%C A350586 In both cases, C, D mean cyclic, dihedral groups of the stated order; the symbols ~ and : mean isomorphic to and semidirect product respectively.
%C A350586 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
%H A350586 Charles R Greathouse IV, <a href="/A350586/b350586.txt">Table of n, a(n) for n = 1..10000</a>
%H A350586 J.H. Conway, H. Dietrich, and E.A. O’Brien, <a href="https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting Groups: Gnus, Moas, and other Exotica</a>, The Mathematical Intelligencer 30, 6-15 (2008), doi:10.1007/BF02985731.
%e A350586 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.
%e A350586 The smallest odd term is 21, the two corresponding groups are C_21 and semi-direct product C_7 : C_3.
%e A350586 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.
%o A350586 (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
%o A350586 (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
%o A350586 (Python)
%o A350586 from sympy import factorint
%o A350586 def is_ok(m):
%o A350586     f = factorint(m)
%o A350586     if any(e > 1 for e in f.values()): return False # m must be squarefree
%o A350586     return sum(q % p == 1 for p in f for q in f) == 1 # _David Radcliffe_, Jul 30 2025
%Y A350586 Equals A054395 \ A350322.
%Y A350586 Subsequence of A060650 and of A005117.
%Y A350586 Cf. A000001, A064899.
%K A350586 nonn
%O A350586 1,1
%A A350586 _Bernard Schott_, Jan 07 2022
%E A350586 More terms from _Jinyuan Wang_, Jan 08 2022