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.

4, 7, 10, 13, 19, 22, 25, 31, 34, 37, 43, 46, 55, 58, 61, 67, 73, 79, 82, 85, 94, 97, 103, 106, 109, 115, 118, 121, 127, 139, 142, 145, 151, 157, 163, 166, 178, 181, 187, 193, 199, 202, 205, 211, 214, 223, 226, 229, 235, 241, 253, 262, 265, 271, 274, 277, 283, 289, 295, 298

Offset: 1

Author

Gleb Ivanov, Oct 03 2021

Keywords

• nonn

Comments

Factorization in S_3 is not unique; for example, 220 = 4 * 55 = 10 * 22.

Crossrefs

Cf. A016777, A057948, A054520.

Programs

Mathematica

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}]]]]

PARI

isok(m) = ((m % 3)==1) && (#select(x->((x%3)==1), divisors(m)) == 2); \\ Michel Marcus, Oct 06 2021

Python

nn = 300
s = [True]*((nn)//3 + 1)
for i in range(4, nn, 3):
    if s[(i-1)//3]:
        for t in range(4, (nn)//i, 3):
            s[(i*t-1)//3] = False
print([3*i + 1 for i in range(1, (nn + 3)//3) if s[i]])