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

1,1

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).

Links

Table of n, a(n) for n=1..50.

Mathematica

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

Prog

(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, ", ")))

Crossrefs

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

Sequence in context: A069455 A211549 A118637 * A139883 A060834 A060844

Adjacent sequences: A096160 A096161 A096162 * A096164 A096165 A096166

Keyword

nonn

Author

Rick L. Shepherd, Jun 18 2004

Status

approved