A231119 Least positive k such that n * k^k + 1 is a prime, or 0 if no such k exists.

1, 1, 2, 1, 102414, 1, 2, 17, 2, 1, 36, 1, 2, 3, 2, 1, 210, 1, 20, 3, 990, 1, 6, 2, 2, 6, 2, 1

Offset: 1

Author

Alex Ratushnyak, Nov 04 2013

Keywords

• nonn
• hard
• more

Comments

The number a(5) is conjectured to be zero. Four days of computation have shown that all numbers 5*k^k+1 are composite for k = 1..22733. - T. D. Noe, Nov 11 2013

The sum of 1/log(n*k^k) diverges slowly for every n so normal heuristics predict infinitely many primes in each case, including n=5. - Jens Kruse Andersen, Jun 16 2014

a(5) > 100000 or a(5) = 0. a(29) > 100000 or a(29) = 0. - Jason Yuen, Jan 06 2025

a(5) = 102414 . - Phillip Poplin, May 27 2025

a(29) > 150000 or a(29) = 0. - Phillip Poplin, May 27 2025

Crossrefs

Cf. A035093, A057217, A175763, A228175.

Programs

Java

import java.math.BigInteger; public class A231119 { public static void main (String[] args) { for (int n = 1; n < 3333; n++) { BigInteger nn = BigInteger.valueOf(n); for (int k=1; k<10000; k++) { BigInteger p = nn.multiply(BigInteger.valueOf(k).pow(k)).add(BigInteger.ONE); if (p.isProbablePrime(80)) { System.out.printf("%d, ", k); break; } else System.out.printf("."); } } } }

Extensions

a(5) from Phillip Poplin, May 27 2025