A122735 Smallest prime of the form (n^k - k^n) for k > 1, or 1 if no such prime exists.

1, 7, 17, 1, 6102977801, 162287, 79792265017612001, 8375575711, 2486784401

Offset: 1

Author

Alexander Adamchuk, Sep 24 2006, corrected Mar 03 2007

Keywords

• nonn

Comments

a(10) = 10^273 - 273^10 is too large to include.

a(16) = 1 because primes of the form (16^k - k^16) do not exist, since 16^k - k^16 = (4^k - k^4)(4^k + k^4).

The corresponding numbers k such that a(n) = (n^k - k^n) are listed in A128355, where k = 0 corresponds to a(n) = 1.

Currently a(n) is not known for n = {17, 18, 22, 25, 26, 27, 28, ...}.

Examples

			a(1) = 1 because (1^k - k^1) = (1 - k) < 0 for k > 1.
a(2) = 7 because 2^5 - 5^2 = 7 is prime, but (2^k - k^2) is not prime for 1 < k < 5, (2^2 - 2^2) = 0, (2^3 - 3^2) = -1, (2^4 - 4^2) = 0.
a(4) = 1 because no prime of the form (4^k - k^4) exists; 4^k - k^4 = (2^k - k^2)*(2^k + k^2).
a(12) = 83695120256591 = 12^13 - 13^12 = A024152(A122003(2)).
		

Crossrefs

Cf. A024152, A122003.

Cf. A128355.