cp's OEIS Frontend

Next term, if it exists, is >4800. - Ryan Propper, Jan 02 2007

From Farideh Firoozbakht, Oct 21 2009: (Start)

Numbers corresponding to a(19)-a(24) are probable primes.

There is no further term up to 8300. (End)

a(n)=2 iff n is odd. If n is even then every prime factor of n+1 is a solution of the equation (n^x + 1) mod x = 0, and if n is odd, the smallest prime factor of n+1 (2) is a solution of (n^x + 1) mod x = 0, so for each n, a(n) is not greater than the smallest prime factor of n+1.

Conjecture 1: All terms of this sequence are primes. We know if n is odd a(n) is the smallest prime factor of n+1.

Conjecture 2: For each n, a(n) is the smallest prime factor of n+1 or a(n)=A020639(n+1).

From Charlie Neder, Jun 16 2019: (Start)

Theorem: a(n) = A020639(n+1).

Proof: If a(n) is composite (kp, say) then n^(kp) == -1 (mod p), but then n^k is also congruent to -1 (mod p) by Fermat's little theorem, contradicting the assumption that a(n) was minimal. Thus, a(n) must be prime, and using Fermat's little theorem again shows that n^p == -1 (mod p) iff n == -1 (mod p), and A020639(n+1) gives the least p such that this is the case. (End)

The theorem plus the conjecture 2 in A092028 imply a(n) = A092028(n+2). - R. J. Mathar, Mar 21 2023

Some larger terms: 4465786944074559659, 1440261542571735083956640176981881665928575750093930787551969