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, 240, 252, 254, 255, 266, 268, 288, 291, 294, 300, 310, 320, 328, 336, 340, 348, 360, 362, 364

Offset

1,1

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.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

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.

Prog

(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, ", "); ); );
(Python)
from itertools import count, combinations, product, islice
from sympy import primefactors, primerange, reduced_totient
def A351899_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        f, t = set(primefactors(k)), reduced_totient(k)
        plist = [p for p in primerange(k) if p not in f]
        for p, q in combinations(plist, 2):
            for i, j in product(range(t+1), repeat=2):
                r = pow(p, i, k)*pow(q, j, k)%k
                if r !=1 and r%p and r%q:
                    break
            else:
                yield k
                break
A351899_list = list(islice(A351899_gen(), 63)) # Chai Wah Wu, Feb 23 2026

Crossrefs

Cf. A306746.

Sequence in context: A044830 A033002 A244021 * A313818 A313819 A313820

Adjacent sequences: A351896 A351897 A351898 * A351900 A351901 A351902

Keyword

nonn

Author

Craig J. Beisel, Feb 24 2022

Extensions

Duplicate term 234 removed by Chai Wah Wu, Feb 23 2026

Status

approved