cp's OEIS Frontend

Numbers of the form k^n*(k^n+1) + 1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even k^(4*n) + k^(2*n) + 1 = (k^(2*n)+1)^2 - (k^n)^2 = (k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3 is k^n*(k^n+1) + 1 always composite?

Phi[3^(n+1),k] = k^(3^n)*(k^(3^n)+1)+1. When m <> 3^n in k^m*(k^m+1)+1, Phi[3m,k] < k^m*(k^m+1)+1 and is a divisor of it. - Lei Zhou, Feb 09 2012

The prime number corresponding to the 10th term is a 587458-digit number. - Lei Zhou, Jul 04 2014

x^(2*k) + x^k + 1 = (x^(3*k) - 1)/(x^k - 1) is the product over n dividing 3k but not dividing k of cyclotomic polynomials Phi(n). If k is a power of 3, n = 3k is the only such divisor and we have a single irreducible cyclotomic polynomial Phi(3k). Otherwise we have the product of more than one polynomial, with integer values > 1 for integer x > 1, and thus always composite numbers. - Martin Becker, Jun 22 2021

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

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 sequence is a subset of A205506 with only m=3^n, which is similar to A153438.

Search limits: a(10) > 35000, a(11) > 3500.

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 sequence is a subset of A205506 with only m=6^n.

Numbers of this form are Generalized unique primes. a(6) generates a 306477-digit prime.