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A280255 Numbers k such that tau(k^(k+1)) is a prime.

%I A280255 #24 Sep 03 2024 12:09:23
%S A280255 3,4,5,11,17,25,29,41,49,59,71,101,107,125,137,149,179,191,197,227,
%T A280255 239,269,281,311,343,347,419,431,461,521,529,569,599,617,641,659,809,
%U A280255 821,827,857,881,1019,1031,1049,1061,1091,1151,1229,1277,1289,1301,1319
%N A280255 Numbers k such that tau(k^(k+1)) is a prime.
%C A280255 tau(k) is the number of positive divisors of k (A000005).
%C A280255 Numbers k such that A000005(A007778(k)) is a prime.
%C A280255 Lesser of twin primes (A001359) are terms. If p is lesser of twin primes then tau(p^(p+1)) = p + 2 (see A006512).
%C A280255 Sequence of composite terms c: 4, 25, 49, 125, 343, 529, 1369, ...; (tau(c^(c+1)): 11, 53, 101, 379, 1033, 1061, 2741, ...).
%C A280255 Numbers of the form p^k where p is prime and 1 + k * (1 + p^k) is prime. - _Robert Israel_, Sep 02 2024
%H A280255 Robert Israel, <a href="/A280255/b280255.txt">Table of n, a(n) for n = 1..10000</a>
%e A280255 tau(4^5) = tau(1024) = 11 (prime).
%p A280255 N:= 10000: # for terms <= N
%p A280255 P:= select(isprime,[2,seq(i,i=3..N,2)]):
%p A280255 R:= {}:
%p A280255 for p in P do
%p A280255   Qs:= select(q -> isprime(1 + q + q*p^q), {$1..ilog[p](N)});
%p A280255   R:= R union map(q -> p^q, Qs)
%p A280255 od:
%p A280255 sort(convert(R,list)); # _Robert Israel_, Sep 02 2024
%t A280255 Select[Range[1319], PrimeQ@DivisorSigma[0, #^(# + 1)] &] (* _Giovanni Resta_, Mar 07 2017 *)
%o A280255 (Magma) [n: n in [1..500] | IsPrime(NumberOfDivisors(n^(n+1)))];
%o A280255 (PARI) isok(n) = isprime(numdiv(n^(n+1))); \\ _Michel Marcus_, Mar 07 2017
%Y A280255 Cf. A000005, A001359, A006512, A007778, A280256, A280257.
%K A280255 nonn
%O A280255 1,1
%A A280255 _Jaroslav Krizek_, Mar 07 2017