login
The odd semiprime numbers (A046315) which are orders of a non-Abelian group.
1

%I #31 Feb 08 2021 23:22:03

%S 21,39,55,57,93,111,129,155,183,201,203,205,219,237,253,291,301,305,

%T 309,327,355,381,417,453,471,489,497,505,543,579,597,633,655,669,687,

%U 689,723,737,755,791,813,831,849,889,905,921,939,955,979,993,1011,1027,1047

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

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

%C The corresponding non-Abelian groups are the semidirect products of Z/qZ and Z/pZ. - _Bernard Schott_, May 16 2020

%H Jinyuan Wang, <a href="/A221048/b221048.txt">Table of n, a(n) for n = 1..1000</a>

%H S. K. Berberian, <a href="http://www.jstor.org/stable/2306479">Non-Abelian Groups of Order pq, American Mathematical Monthly</a>, Vol. 60, No. 1, Jan. 1953, 37-40.

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

%o (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

%o (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

%Y Intersection of A046315 and A060652.

%K nonn

%O 1,1

%A _David Brown_, Apr 14 2013

%E More terms from _Jinyuan Wang_, May 16 2020