A341670 a(n) is conjecturally the largest prime p such that, for every prime q > p, q^n - 1 has more divisors than does p^n - 1, or -1 if no such prime p exists.

5, 73, 5, 101, 3, 167, 5, 71, 3, 43, 5, 167, 5, 73, 3, 19, 2, 17, 2, 19, 3, 23, 2, 71, 2, 7, 2, 29, -1, 13, 2, 11, 2, 7, 2, 13, 2, 3, 2, 11, 2, 7, -1, 23, 2, 17, 5, 17, 5, 7, -1, 7, -1, 11, -1, 5, 2, 7, 2, 11, 2, 2, 3, 3, 2, 5, 2, 2, 2, 3, 3, 11, -1, 2, -1, 7

Offset: 1

Author

Jon E. Schoenfield, Feb 26 2021

Keywords

• sign

Comments

Conjecture: a(n) is also the largest prime p for which there exists no prime q > p such that p^n - 1 and q^n - 1 have the same number of divisors.

Examples

			It is conjectured that there are infinitely many primes q such that q-1 has exactly A309906(1)=4 divisors (see also A005385), and p=5 is the largest prime p such that p-1 has fewer than 4 divisors, so a(1)=5.
Similarly, there appear to be infinitely many primes q such that q^2 - 1 has exactly A309906(2)=32 divisors (e.g., primes q such that, of the two factors q-1 and q+1 of q^2 - 1, one is twice a prime > 3 and the other is 12 times a prime > 3), and p=73 is the largest prime p such that p^2 - 1 has fewer than 32 divisors (see A341655), so a(2)=73.
a(4)=101 is the largest prime p such that p^4 - 1 has fewer than 160 divisors, and there are conjecturally infinitely many primes q such that q^4 - 1 has exactly A309906(4)=160 divisors.
a(29)=-1 because there are conjecturally infinitely many primes q such that q^29 - 1 has exactly A309906(29)=8 divisors, and there exists no prime p such that p^29 - 1 has fewer than 8 divisors.
		

Crossrefs

Cf. A000005, A000040, A005385, A309906, A341655, A341656, A341657, A341658, A341659.