A351899 Integers k for which there exist two distinct prime nondivisors p, q < k such that, for all i, j >= 0, p^i*q^j mod k is either 1 or is divisible by p or q.

5, 10, 16, 18, 19, 20, 21, 22, 38, 48, 50, 51, 54, 60, 61, 67, 75, 77, 78, 80, 85, 90, 91, 98, 100, 108, 120, 122, 126, 127, 134, 147, 150, 154, 160, 170, 182, 189, 201, 204, 210, 217, 234, 234, 240, 252, 254, 255, 266, 268, 288, 291, 294, 300, 310, 320, 328, 336, 340, 348, 360, 362, 364

Offset: 1

Author

Craig J. Beisel, Feb 24 2022

Keywords

• nonn

Comments

Conjecture: The prime nondivisors p and q are elements of the reduced residue system consisting of the totatives of k. Assume a triple (k,p,q) which satisfies the definition. If P and Q are the two subgroups generated by p and q respectively and p < q then P >= Q.

Examples

			For k = 20 and p, q = (3,7), p^i*q^j mod k can only take on the values 1, 3, 7, 9 which, other than 1, are all divisible by 3 or 7, so 20 is a term.
		

Crossrefs

Cf. A306746.

Programs

PARI

for(k=1, 364, test2=0; forprime(p=2, k-1, forprime(q=p+1, k-2, if(gcd(p, k)==1 && gcd(q, k)==1, test=0; for(i=0, eulerphi(k), for(j=0, eulerphi(k), if(p^i*q^j % k >1 && gcd(p^i*q^j % k, p)==1 && gcd(p^i*q^j % k, q)==1, test=1; ); if(test==1, break(2); ); ); ); if(test==0, test2=1; ); ); ); ); if(test2==1, print1(k, ", "); ); );