A246121 Least k such that k^(6^n)*(k^(6^n) - 1) + 1 is prime.

2, 3, 88, 28, 688, 7003, 1925

Offset: 0

Author

Serge Batalov, Aug 14 2014

Keywords

• nonn
• more
• hard

Comments

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.

Examples

			When k = 88, k^72 - k^36 + 1 is prime. Since this isn't prime for k < 88, a(2) = 88.
		

Links

• C. Caldwell, Generalized unique primes

Crossrefs

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246119, A246120.

Programs

PARI

a(n)=k=1; while(!ispseudoprime(k^(6^n)*(k^(6^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++)

Formula

a(n) = A085398(6^(n+1)). - Jinyuan Wang, Jan 01 2023

Extensions

a(6) from Serge Batalov, Aug 15 2014