cp's OEIS Frontend

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

%I A300217 #23 Apr 06 2025 19:55:58
%S A300217 3,4,5,6,8,10,12,17,32,34,40,48,60,85,128,136,160,170,192,204,240,
%T A300217 1285,2048,2056,2176,2560,2570,2720,3072,3084,3264,3840,4080,4369,
%U A300217 8192,8224,8704,8738,10240,10280,10880,12288,12336,13056,15360,15420,16320,65537
%N A300217 Numbers k such that tau(phi(k)) is a prime.
%C A300217 Numbers k such that A062821(k) = A000005(A000010(k)) is a prime.
%C A300217 Supersequence of A062514.
%C A300217 From _Robert Israel_, Mar 18 2018: (Start)
%C A300217 Numbers k such that A000010(k) = 2^(p-1) where p is prime.
%C A300217 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.
%C A300217 (End)
%C A300217 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
%H A300217 Jaroslav Krizek, <a href="/A300217/b300217.txt">Table of n, a(n) for n = 1..100</a>
%e A300217 17 is a term because phi(17) = 16, tau(16) = 5 (prime).
%p A300217 select(isprime @ numtheory:-tau @ numtheory:-phi, [$1..10^5]); # _Robert Israel_, Mar 18 2018
%t A300217 Select[Range[2^16 + 1], PrimeQ@ DivisorSigma[0, EulerPhi@ #] &] (* _Michael De Vlieger_, Mar 01 2018 *)
%o A300217 (Magma) [n: n in[1..10^6] | IsPrime(NumberOfDivisors(EulerPhi(n)))];
%o A300217 (PARI) isok(k) = isprime(numdiv(eulerphi(k))); \\ _Altug Alkan_, Mar 04 2018
%Y A300217 Cf. A000005, A000010, A062514, A062821, A019434, A249759.
%K A300217 nonn
%O A300217 1,1
%A A300217 _Jaroslav Krizek_, Feb 28 2018