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A348156 S_3-primes: let S_3 = {1,4,7,...,3i+1,...}; then an S_3-prime is in S_3 but is not divisible by any elements of S_3 except for itself and 1.

%I A348156 #25 Dec 04 2021 12:52:40
%S A348156 4,7,10,13,19,22,25,31,34,37,43,46,55,58,61,67,73,79,82,85,94,97,103,
%T A348156 106,109,115,118,121,127,139,142,145,151,157,163,166,178,181,187,193,
%U A348156 199,202,205,211,214,223,226,229,235,241,253,262,265,271,274,277,283,289,295,298
%N A348156 S_3-primes: let S_3 = {1,4,7,...,3i+1,...}; then an S_3-prime is in S_3 but is not divisible by any elements of S_3 except for itself and 1.
%C A348156 Factorization in S_3 is not unique; for example, 220 = 4 * 55 = 10 * 22.
%t A348156 nn = 100; Complement[Table[3 k + 1, {k, 1, nn}], Union[Flatten[ Table[Table[(3 k + 1) (3 j + 1), {k, 1, j}], {j, 1, nn}]]]]
%o A348156 (PARI) isok(m) = ((m % 3)==1) && (#select(x->((x%3)==1), divisors(m)) == 2); \\ _Michel Marcus_, Oct 06 2021
%o A348156 (Python)
%o A348156 nn = 300
%o A348156 s = [True]*((nn)//3 + 1)
%o A348156 for i in range(4, nn, 3):
%o A348156     if s[(i-1)//3]:
%o A348156         for t in range(4, (nn)//i, 3):
%o A348156             s[(i*t-1)//3] = False
%o A348156 print([3*i + 1 for i in range(1, (nn + 3)//3) if s[i]])
%Y A348156 Cf. A016777, A057948, A054520.
%K A348156 nonn
%O A348156 1,1
%A A348156 _Gleb Ivanov_, Oct 03 2021