A096163 Primes p of the form qrs + 1 where q, r and s are distinct primes.

31, 43, 67, 71, 79, 103, 131, 139, 191, 223, 239, 283, 311, 367, 419, 431, 439, 443, 499, 599, 607, 619, 643, 647, 659, 683, 743, 787, 823, 827, 907, 947, 971, 1031, 1039, 1087, 1091, 1103, 1163, 1223, 1259, 1399, 1427, 1447, 1499, 1511, 1543, 1559, 1571, 1579

Offset: 1

Author

Rick L. Shepherd, Jun 18 2004

Keywords

• nonn

Comments

Each composite number qrs = a(n)-1 is a squarefree 3-almost prime. This sequence is a subsequence of A078330 which, besides having 3 as its first term, first differs by including 2311 = 2*3*5*7*11 + 1 (a squarefree 5-almost prime plus 1).

Crossrefs

Cf. A078330 (primes p with mu(p-1) = -1).

Programs

Mathematica

With[{nn=50},Take[Union[Select[Times@@@Subsets[Prime[Range[2nn]],{3}]+1,PrimeQ]],nn]] (* Harvey P. Dale, Jun 06 2021 *)

PARI

/* Here are five equivalent PARI programs */ forprime(p=2,2400, if(moebius(p-1)==-1 && omega(p-1)==3, print1(p,","))) forprime(p=2,2400, if(moebius(p-1)==-1 && bigomega(p-1)==3, print1(p,","))) forprime(p=2,2400, if(bigomega(p-1)==3 && omega(p-1)==3, print1(p,","))) forprime(p=2,2400, if(omega(p-1)==3 && issquarefree(p-1), print1(p,","))) forprime(p=2,2400, if(bigomega(p-1)==3 && issquarefree(p-1), print1(p,",")))