A280255 Numbers k such that tau(k^(k+1)) is a prime.

3, 4, 5, 11, 17, 25, 29, 41, 49, 59, 71, 101, 107, 125, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 343, 347, 419, 431, 461, 521, 529, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319

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(A007778(k)) is a prime.

Lesser of twin primes (A001359) are terms. If p is lesser of twin primes then tau(p^(p+1)) = p + 2 (see A006512).

Sequence of composite terms c: 4, 25, 49, 125, 343, 529, 1369, ...; (tau(c^(c+1)): 11, 53, 101, 379, 1033, 1061, 2741, ...).

Numbers of the form p^k where p is prime and 1 + k * (1 + p^k) is prime. - Robert Israel, Sep 02 2024

Examples

			tau(4^5) = tau(1024) = 11 (prime).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001359, A006512, A007778, A280256, A280257.

Programs

Magma

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

Maple

N:= 10000: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
R:= {}:
for p in P do
  Qs:= select(q -> isprime(1 + q + q*p^q), {$1..ilog[p](N)});
  R:= R union map(q -> p^q, Qs)
od:
sort(convert(R,list)); # Robert Israel, Sep 02 2024

Mathematica

Select[Range[1319], PrimeQ@DivisorSigma[0, #^(# + 1)] &] (* Giovanni Resta, Mar 07 2017 *)

PARI

isok(n) = isprime(numdiv(n^(n+1))); \\ Michel Marcus, Mar 07 2017