A243147 Least number k such that n^k + k^n is prime or 0 if no such k exists.

1, 1, 2, 1, 24, 1, 54, 69, 2, 1, 3100, 1

Offset: 1

Author

Derek Orr, May 30 2014

Keywords

• nonn
• hard
• more

Comments

More terms given in links.

a(n) = 1 if and only if n + 1 is prime. Thus there are infinitely many nonzero entries.

For n in A016767, a(n) = 0 since n^k + k^n is factorable and will never be prime. Thus there are infinitely many zero entries.

If a(i) = j then a(j) <= i for all i and j not equal to 0.

a(n) and n must have opposite parity. If n is odd/even, a(n) must be even/odd, respectively.

Further, gcd(n, a(n)) = 1 for all n.

Examples

			3^1 + 1^3 = 4 is not prime. 3^2 + 2^3 = 17 is prime. So a(3) = 2.
		

Links

• H. Lifchitz and R. Lifchitz, PRP Top Records. Search for x^y+y^x
• Derek Orr, Table for n, a(n) for n = 1..100 (unknown k-values are marked "unknown")

Crossrefs

Cf. A016767.

Programs

PARI

a(n)=if(ispower(n)&&ispower(n)%3==0&&n%3==0,return(0));k=1;while(!ispseudoprime(n^k+k^n),k++);return(k)
vector(12, n, a(n))