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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.

%I A341670 #20 Feb 27 2021 21:47:15
%S A341670 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,
%T A341670 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,
%U A341670 -1,5,2,7,2,11,2,2,3,3,2,5,2,2,2,3,3,11,-1,2,-1,7
%N 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.
%C A341670 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.
%e A341670 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.
%e A341670 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.
%e A341670 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.
%e A341670 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.
%Y A341670 Cf. A000005, A000040, A005385, A309906, A341655, A341656, A341657, A341658, A341659.
%K A341670 sign
%O A341670 1,1
%A A341670 _Jon E. Schoenfield_, Feb 26 2021