A300217 Numbers k such that tau(phi(k)) is a prime.

3, 4, 5, 6, 8, 10, 12, 17, 32, 34, 40, 48, 60, 85, 128, 136, 160, 170, 192, 204, 240, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 4369, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 15360, 15420, 16320, 65537

Offset: 1

Author

Jaroslav Krizek, Feb 28 2018

Keywords

• nonn

Comments

Numbers k such that A062821(k) = A000005(A000010(k)) is a prime.

Supersequence of A062514.

From Robert Israel, Mar 18 2018: (Start)

Numbers k such that A000010(k) = 2^(p-1) where p is prime.

Numbers of the form 2^m*Product_{i=1..k} (2^(2^(e_i))+1) where 2^(2^(e_i)+1) are distinct Fermat primes (A019434) and m + 1 + Sum_i 2^(e_i) is prime. In particular the prime terms are A249759.

(End)

According to a comment in A009087, if the sum of divisors is prime, then the number of divisors is also prime. - Michael B. Porter, Mar 23 2018

Examples

			17 is a term because phi(17) = 16, tau(16) = 5 (prime).
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000005, A000010, A062514, A062821, A019434, A249759.

Programs

Magma

[n: n in[1..10^6] | IsPrime(NumberOfDivisors(EulerPhi(n)))];

Maple

select(isprime @ numtheory:-tau @ numtheory:-phi, [$1..10^5]); # Robert Israel, Mar 18 2018

Mathematica

Select[Range[2^16 + 1], PrimeQ@ DivisorSigma[0, EulerPhi@ #] &] (* Michael De Vlieger, Mar 01 2018 *)

PARI

isok(k) = isprime(numdiv(eulerphi(k))); \\ Altug Alkan, Mar 04 2018