%I #9 Apr 09 2024 03:26:48

%S 1,2,3,5,7,9,11,13,17,19,23,25,28,29,31,37,40,41,43,45,47,49,53,59,61,

%T 67,71,73,79,81,83,89,97,101,103,107,109,113,121,127,131,137,139,149,

%U 151,153,157,163,167,169,173,179,181,191,193,197,199,211,223,225,227

%N Numbers m such that m^k has m divisors for some k >= 1.

%C All primes p are in this sequence, since p^(p-1) has p divisors.

%C For all odd semiprimes s, s^2 is in this sequence, since s^((s-1)/2) has s divisors.

%H David W. Wilson, <a href="/A180934/b180934.txt">Table of n, a(n) for n=1..10000</a>

%e 11^10 has 11 divisors, so 11 is in the sequence.

%e 225^7 has 225 divisors, so 225 is in the sequence.

%t q[n_] := Module[{e = FactorInteger[n][[;; , 2]], k = 1}, While[n > Times @@ (k*e + 1), k++]; n == Times @@ (k*e + 1)]; q[1] = True; Select[Range[250], q] (* _Amiram Eldar_, Apr 09 2024 *)

%Y A000005(m^k) = m for some k >= 1.

%Y A180935 gives the corresponding k.

%Y Cf. A036878, A046315.

%K nonn

%O 1,2

%A _David W. Wilson_, Sep 26 2010