A280256 - OEIS

%I #14 Jan 16 2023 21:41:24

%S 2,9,6561,25937424601,1853020188851841,58149737003040059690390169,

%T 54116956037952111668959660849,

%U 2787593149816327892691964784081045188247552,2465034704958067503996131453373943813074726512397600969

%N Numbers k such that tau(k^k) is a prime.

%C tau(k) is the number of positive divisors of k (A000005).

%C Numbers k such that A000005(A000312(k)) = A062319(k) is a prime.

%C Corresponding values of primes: 3, 19, 52489, ...

%C All the terms are prime powers.

%H Giovanni Resta, <a href="/A280256/b280256.txt">Table of n, a(n) for n = 1..200</a>

%e tau(9^9) = tau(387420489) = 19 (prime).

%t mx = 10^200; Union@ Flatten@ Reap[ Sow[2^ Select[ Range@ Log2[mx], PrimeQ[1 + # 2^#] &]]; Do[ If[ PrimeQ[1 + q p^q], Sow[p^q]], {p, Prime@ Range@ PrimePi@ 34}, {q, 2, Log[p, mx], 2}]; Do[ Sow@ (Select[ Prime@ Range[2, PrimePi[ mx^(1/e)]], PrimeQ[1 + e #^e] &]^e), {e, 34, Floor@Log[31, mx], 2}]][[2, 1]] (* all the 231 terms < 10^200, _Giovanni Resta_, Mar 07 2017 *)

%o (Magma) [n: n in [1..500] | IsPrime(NumberOfDivisors(n^n))]

%o (PARI) isok(n) = isprime(numdiv(n^n)); \\ _Michel Marcus_, Mar 07 2017

%Y Cf. A000005, A000312, A062319, A280255, A280257.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Mar 07 2017

%E a(4)-a(9) from _Giovanni Resta_, Mar 07 2017