A221048 The odd semiprime numbers (A046315) which are orders of a non-Abelian group.

21, 39, 55, 57, 93, 111, 129, 155, 183, 201, 203, 205, 219, 237, 253, 291, 301, 305, 309, 327, 355, 381, 417, 453, 471, 489, 497, 505, 543, 579, 597, 633, 655, 669, 687, 689, 723, 737, 755, 791, 813, 831, 849, 889, 905, 921, 939, 955, 979, 993, 1011, 1027, 1047

Offset: 1

Author

David Brown, Apr 14 2013

Keywords

• nonn

Comments

Numbers of the form pq where p,q are odd primes, p

The corresponding non-Abelian groups are the semidirect products of Z/qZ and Z/pZ. - Bernard Schott, May 16 2020

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• S. K. Berberian, Non-Abelian Groups of Order pq, American Mathematical Monthly, Vol. 60, No. 1, Jan. 1953, 37-40.

Crossrefs

Intersection of A046315 and A060652.

Programs

Mathematica

Select[1 + 2*Range[500], (f = FactorInteger[#]; Last /@ f == {1, 1} && Mod @@ Reverse[First /@ f] == 1) &] (* Giovanni Resta, Apr 14 2013 *)

PARI

lista(nn) = {forstep(n=1, nn, 2, my(f=factor(n)); if ((#f~ == 2) && (vecmax(f[,2]) == 1) && ((f[2,1] % f[1,1]) == 1), print1(n, ", ")););} \\ Michel Marcus, Sep 28 2017

PARI

list(lim)=my(v=List()); if(lim<9, return([])); forprime(p=3,sqrtint(((lim\=1)-1)\2), forprimestep(q=2*p+1,lim,2*p, listput(v, p*q))); Set(v) \\ Charles R Greathouse IV, Feb 08 2021

Extensions

More terms from Jinyuan Wang, May 16 2020

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