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

%I A280257 #18 Jan 16 2023 21:41:29
%S A280257 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,
%T A280257 71,73,79,83,89,97,101,103,107,109,113,121,125,127,131,137,139,149,
%U A280257 151,157,163,167,169,173,179,181,191,193,197,199,211,223,227,229
%N A280257 Numbers k such that tau(k^(k-1)) is a prime.
%C A280257 tau(k) is the number of positive divisors of k (A000005).
%C A280257 Numbers k such that A000005(A000169(k)) is a prime.
%C A280257 All primes (A000040) are terms. If p is prime then tau(p^(p-1)) = p.
%C A280257 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, ...).
%C A280257 All terms are powers of primes (A000961). - _Robert Israel_, Mar 07 2017
%H A280257 Robert Israel, <a href="/A280257/b280257.txt">Table of n, a(n) for n = 1..10000</a>
%F A280257 a(n) ~ n log n. - _Charles R Greathouse IV_, Mar 07 2017
%e A280257 tau(4^3) = tau(64) = 7 (prime).
%p A280257 N:= 5000: # to get all terms <= N
%p A280257 Primes:= select(isprime, {2,seq(i,i=3..N,2)}):
%p A280257 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
%t A280257 Select[Range@ 230, PrimeQ@ DivisorSigma[0, #^(# - 1)] &] (* _Michael De Vlieger_, Mar 07 2017 *)
%o A280257 (Magma) [n: n in [1..100] | IsPrime(NumberOfDivisors(n^(n-1)))]
%o A280257 (PARI) isok(n) = isprime(numdiv(n^(n-1))); \\ _Michel Marcus_, Mar 07 2017
%o A280257 (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
%Y A280257 Cf. A000005, A000040, A000169, A000961, A280255, A280256.
%K A280257 nonn,easy
%O A280257 1,1
%A A280257 _Jaroslav Krizek_, Mar 07 2017