Numbers of the form k^m*(k^m-1)+1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of
A205506 with only m=2^n.
Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.
The numbers of this form are Generalized Unique primes (see Links section).
a(16)=96873 corresponds to a prime with 653552 decimal digits.
The search for a(17) which corresponds to a 1385044-decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.
a(18) <= 712012 corresponds to a prime with 3068389 decimal digits. -
Serge Batalov, Jan 15 2018
a(19) <= 123447 corresponds to a prime with 5338805 decimal digits. -
Serge Batalov, Jan 15 2018
a(20) <= 465859 corresponds to a prime with 11887192 decimal digits (not all lower candidates have been checked). This is the largest known non-Mersenne prime at the time of its discovery. -
Serge Batalov, May 31 2023
