A343828 Numbers which are the product of two S-primes (A057948) in exactly three ways.

4389, 5313, 7161, 9177, 9933, 10857, 12369, 13629, 14421, 14973, 15477, 16401, 17157, 18249, 18753, 19173, 19437, 20769, 22701, 23529, 23541, 23793, 24717, 26733, 26961, 27993, 28329, 28497, 29337, 29469, 30261, 30597, 31521, 32109, 32361, 32637, 33117, 33649

Offset: 1

Author

Zachary DeStefano, Apr 30 2021

Keywords

• nonn

Comments

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways.

An S-prime is either a prime of the form 4k+1 or a semiprime of the form (4k+3)*(4m+3). That means the maximum number of prime factors that a number factorizable into two S-primes can have is four (all 4k + 3), and those can be combined into S-primes in at most three distinct ways. - Gleb Ivanov, Dec 07 2021

Examples

			9177 = 21*437 = 57*161 = 69*133 which are all S-primes (A057948), and admits no other S-Prime factorizations.
4389 = (3*7)*(11*19) = (3*11)*(7*19) = (3*19)*(7*11); 3,7,11,19 are the smallest primes of the form 4k + 3.
		

Links

• Zachary DeStefano, Table of n, a(n) for n = 1..1014

Crossrefs

Cf. A054520, A057948, A057949, A057950.

Exactly one way: A343826. Exactly two ways: A343827.

Programs

PARI

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 3; \\ Michel Marcus, May 01 2021

Formula

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021