A246120 Least k such that k^(3^n)*(k^(3^n) - 1) + 1 is prime.

2, 6, 7, 93, 15, 372, 421, 759, 7426, 9087

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=3^n, which is similar to A153438.

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

Examples

			When k = 7, k^18 - k^9 + 1 is prime. Since this isn't prime for k < 7, a(2) = 7.
		

Crossrefs

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246119, A246121.

Programs

Mathematica

a246120[n_Integer] := Module[{k = 1},
  While[! PrimeQ[k^(3^n)*(k^(3^n) - 1) + 1], k++]; k]; a246120 /@ Range[0, 9] (* Michael De Vlieger, Aug 15 2014 *)

PARI

a(n)=k=1;while(!ispseudoprime(k^(3^n)*(k^(3^n)-1)+1),k++);k
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Aug 14 2014

Formula

a(n) = A085398(2*3^(n+1)). - Jinyuan Wang, Jan 01 2023