A258429 Primes p such that p - 1 = (tau(p - 1) - 1)^k for some k >= 0, where tau(n) is the number of divisors of n (A000005).

2, 5, 17, 65537

Offset: 1

Author

Jaroslav Krizek, May 29 2015

Keywords

• nonn

Comments

Conjecture: the sequence is finite.

Corresponding values of numbers k: 0, 2, 2, 4, ...

A Fermat prime from A019434 of the form F(n) = 2^(2^n) + 1 is a term if k = 2^n * log(2) / log(2^n) is an integer.

Examples

			65537 (prime) is in the sequence because 65537 - 1 = (tau(65536) - 1)^4 = 16^4.
		

Crossrefs

Cf. A000005, A019434, A219338, A007516, A004249, A249759.

Programs

Magma

[2] cat [n+1: n in [A219338(n)] | IsPrime(n+1)];

Magma

Set(Sort([n: n in[1..1000000], k in [0..100] | IsPrime(n) and (n-1) eq (NumberOfDivisors(n-1) - 1)^k]));

PARI

listp(nn) = {print1(p=2, ", "); forprime(p=5, nn, expo = valuation(x=(p-1), y=(numdiv(p-1)-1)); if (x == y^expo, print1(p, ", ")););} \\ Michel Marcus, Jun 04 2015