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

1,1

Comments

Numbers of the form pq where p,q are odd primes, p<q and q is congruent to 1 mod 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.

Mathematica

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

Prog

(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

Crossrefs

Intersection of A046315 and A060652.

Sequence in context: A072708 A338330 A102478 * A182297 A338552 A339002

Adjacent sequences: A221045 A221046 A221047 * A221049 A221050 A221051

Keyword

nonn

Author

David Brown, Apr 14 2013

Extensions

More terms from Jinyuan Wang, May 16 2020

Status

approved