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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.

%I A351899 #32 Mar 09 2022 16:40:48
%S A351899 5,10,16,18,19,20,21,22,38,48,50,51,54,60,61,67,75,77,78,80,85,90,91,
%T A351899 98,100,108,120,122,126,127,134,147,150,154,160,170,182,189,201,204,
%U A351899 210,217,234,234,240,252,254,255,266,268,288,291,294,300,310,320,328,336,340,348,360,362,364
%N 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.
%C A351899 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.
%e A351899 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.
%o A351899 (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, ", "); ); );
%Y A351899 Cf. A306746.
%K A351899 nonn
%O A351899 1,1
%A A351899 _Craig J. Beisel_, Feb 24 2022