A215068 Numbers n such that for all divisors d of n, d+1 is either a prime or a perfect power.

1, 2, 3, 4, 6, 7, 8, 12, 16, 24, 31, 48, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Author

Joerg Arndt, Aug 02 2012

Keywords

• nonn
• hard

Comments

Apparently the divisors of 48 (A018261) together with the Mersenne primes (A000668).

Confirmed by Robert Israel, Aug 02 2020: see link.

Next term > 2*10^8.

Links

• Robert Israel, Proof of conjecture in A215068

Crossrefs

Cf. A018261 (divisors of 48), A000668 (Mersenne primes), A001597 (perfect powers).

Programs

Maple

sort([op(numtheory:-divisors(48)), seq(numtheory:-mersenne([i]),i=2..12)]); # Robert Israel, Aug 02 2020

PARI

isA215068(n)=
{
    my(x);
    fordiv (n, d,
        d1 = d + 1;
        if ( isprime(d1) || ispower(d1), next() );
        return(0);
    );
    return(1);
}
for (n=1,10^9, if(isA215068(n), print1(n,", ")));