A280257 Numbers k such that tau(k^(k-1)) is a prime.

2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Author

Jaroslav Krizek, Mar 07 2017

Keywords

• nonn
• easy

Comments

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

Numbers k such that A000005(A000169(k)) is a prime.

All primes (A000040) are terms. If p is prime then tau(p^(p-1)) = p.

Sequence of composite terms c: 4, 9, 16, 27, 49, 64, 121, 125, 169, 289, ...; (tau(c^(c-1)): 7, 17, 61, 79, 97, 379, 241, 373, 337, 577, ...).

All terms are powers of primes (A000961). - Robert Israel, Mar 07 2017

Examples

			tau(4^3) = tau(64) = 7 (prime).
		

Links

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

Crossrefs

Cf. A000005, A000040, A000169, A000961, A280255, A280256.

Programs

Magma

[n: n in [1..100] | IsPrime(NumberOfDivisors(n^(n-1)))]

Maple

N:= 5000: # to get all terms <= N
Primes:= select(isprime, {2,seq(i,i=3..N,2)}):
sort([seq(seq(`if`(isprime(k*(p^k-1)+1),p^k,NULL), k=1..floor(log[p](N))), p=Primes)]); # Robert Israel, Mar 07 2017

Mathematica

Select[Range@ 230, PrimeQ@ DivisorSigma[0, #^(# - 1)] &] (* Michael De Vlieger, Mar 07 2017 *)

PARI

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

PARI

list(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), if(ispseudoprime(e*(p^e-1)+1), listput(v,p^e)))); Set(v) \\ Charles R Greathouse IV, Mar 07 2017

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Mar 07 2017