A130827 Least k >= 1 such that k^n + n is semiprime, or 0 if no such k exists.

3, 2, 1, 3, 1, 7, 3, 1, 1, 11, 2, 7, 1, 1, 7, 3, 5, 23, 4, 1, 1, 3, 2, 1, 1, 21, 14, 11, 12, 7, 16, 1, 1, 1, 26, 37, 1, 1, 4, 21, 6, 31, 4, 25, 1, 71, 14, 1, 10, 1, 10, 371, 36, 1, 3, 1, 1, 185, 2, 43, 1, 49, 104, 1, 18, 205, 70, 1, 2, 33, 38, 541, 1, 105, 8, 1, 24, 395, 30, 3, 1, 71, 20, 1, 1, 1

Offset: 1

Author

Zak Seidov, Aug 18 2007

Keywords

• nonn

Comments

There exist values of n for which k^n + n is never prime (cf. A072883). Do there exist values of n for which k^n + n is never semiprime?

Compare with A361803, the equivalent sequence for k^n - n, where a generalized factorization (effectively a polynomial factorization) into 3 factors is given to show that k^n - n is never semiprime for certain n. - Peter Munn, Jun 19 2023

Examples

			a(1)=3 because 1^1 + 1 = 2 (prime) and 2^1 + 1 = 3 (prime) but 3^1 + 1 = 4 = 2*2 (semiprime).
a(2)=2 because 1^2 + 2 = 3 (prime) but 2^2 + 2 = 6 = 2*3 (semiprime).
a(3)=1 because 1^3 + 3 = 4 = 2*2 (semiprime).
a(4)=3 because 1^4 + 4 = 5 (prime) and 2^4 + 4 = 20 = 2^2 * 5 but 3^4 + 4 = 85 = 5*17 (semiprime).
a(5)=1 because 1^5 + 5 = 6 = 2*3 (semiprime).
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..100.

Crossrefs

Cf. A097792 (n such that x^n+n is reducible), A072883 (least k >= 1 such that k^n+n is prime, or 0 if no such k exists).

Cf. A361803.

Programs

PARI

a(n) = my(k=1); while (bigomega(k^n+n)!=2, k++); k; \\ Michel Marcus, Jun 19 2023

Extensions

More terms from Sean A. Irvine, Oct 20 2009