A228175 Least positive k such that n^n * k^k + 1 is a prime, or 0 if no such k exists.

1, 1, 1, 2, 1, 6, 5, 2, 7, 10, 8

Offset: 0

Author

Alex Ratushnyak, Nov 02 2013

Keywords

• nonn
• hard
• more

Comments

The next terms after the missing a(11) are 31, 58, 4, 596, 3.

a(11) > 20000 or a(11) = 0, a(17) = 4308, a(18) = 1073, a(19) > 20000 or a(19) = 0. - Jason Yuen, May 21 2024

a(11) > 10^5 or a(11) = 0. a(19) > 10^5 or a(19) = 0. - Jason Yuen, Feb 27 2025

Examples

			3^3 * 1 + 1 = 28 is not a prime, 3^3 * 2^2 + 1 = 109 is a prime, so a(3) = 2.
		

Crossrefs

Cf. A000040, A000312, A156223, A231119, A380903.

Programs

Java

import java.math.BigInteger;
public class A228175 {
  public static void main (String[] args) {
    for (int n = 0; n < 333; n++) {
      BigInteger nn = BigInteger.valueOf(n).pow(n);
      int k = 1;
      for (; k<10000; k++) {
        BigInteger kk = BigInteger.valueOf(k).pow(k).multiply(nn).add(BigInteger.ONE);
        if (kk.isProbablePrime(80)) {
          System.out.printf("%d, ", k);
          break;
        }
      }
      if (k==10000) System.out.printf("- ");
    }
  }
}

PARI

A228175(n,L=9e9,s=1)={forstep(k=s+(bittest(n,0)&&n>1&&bittest(s,0)), L, 1+bittest(n,0), ispseudoprime(n^n*k^k+1)&&return(k))} \\ Optional args allow specification of start and limit for search; for odd n > 1, only check even k. - M. F. Hasler, Nov 03 2013

Formula

a(n) = A231119(n^n). - Jason Yuen, Nov 15 2024