A279094 Smallest k such that sigma(k^n) is prime.

2, 2, 4, 2, 25, 2, 59049, 4, 4, 5, 256, 2, 282475249, 243, 4, 2, 729, 2, 1174562876521148458974062689, 8, 64, 16, 25, 1331, 594823321, 16807, 38950081, 151, 361, 2, 470541197898347534873984161, 19902511, 241081, 27, 9, 61, 625, 34271896307633, 73441, 53, 1681

Offset: 1

Author

Jon E. Schoenfield, Mar 11 2017

Keywords

• nonn

Comments

For any number k with two or more distinct prime divisors, the sum of divisors of k^n is composite, so each term is of the form p^j where p is prime and j >= 1, i.e., all terms are prime powers (A246655). Additionally, sigma(k^n) = sigma(p^(j*n)) = (p^(j*n + 1) - 1)/(p - 1) is composite when j*n + 1 is composite, so a(n) must be of the form p^j where j*n + 1 is prime.

Examples

			a(1) = 2 because sigma(1^1) = sigma(1) = 1 (not prime), but sigma(2^1) = sigma(2) = 1 + 2 = 3 (prime).
a(3) = 4 because sigma(1^3) = 1 (not prime), sigma(2^3) = 1 + 2 + 4 + 8 = 15 (composite), sigma(3^3) = 1 + 3 + 9 + 27 = 40 (composite), but sigma(4^3) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 (prime).
a(19) = 1174562876521148458974062689 = 17^22 because sigma((17^22)^19) is prime and sigma(k^19) is not prime for any smaller value of k.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000203, A023194, A055638, A246655.