A280256 Numbers k such that tau(k^k) is a prime.

2, 9, 6561, 25937424601, 1853020188851841, 58149737003040059690390169, 54116956037952111668959660849, 2787593149816327892691964784081045188247552, 2465034704958067503996131453373943813074726512397600969, 285273917723723876056171083405292782327767461712708093041

Offset: 1

Author

Jaroslav Krizek, Mar 07 2017

Keywords

• nonn

Comments

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

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

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

All the terms are prime powers.

Examples

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..200

Crossrefs

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

Programs

Magma

[n: n in [1..500] | IsPrime(NumberOfDivisors(n^n))];

Mathematica

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 *)

PARI

isok(n) = isprime(numdiv(n^n)); \\ Michel Marcus, Mar 07 2017

Extensions

a(4)-a(10) from Giovanni Resta, Mar 07 2017