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A279094 Smallest k such that sigma(k^n) is prime. 5

%I

%S 2,2,4,2,25,2,59049,4,4,5,256,2,282475249,243,4,2,729,2,

%T 1174562876521148458974062689,8,64,16,25,1331,594823321,16807,

%U 38950081,151,361,2,470541197898347534873984161,19902511,241081,27,9,61,625,34271896307633,73441,53,1681

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

%C 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.

%H Jon E. Schoenfield, <a href="/A279094/b279094.txt">Table of n, a(n) for n = 1..200</a>

%e a(1) = 2 because sigma(1^1) = sigma(1) = 1 (not prime), but sigma(2^1) = sigma(2) = 1 + 2 = 3 (prime).

%e 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).

%e 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.

%Y Cf. A000203, A023194, A055638, A246655.

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Mar 11 2017

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)