A180934 Numbers m such that m^k has m divisors for some k >= 1.

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, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 153, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227

Offset: 1

Author

David W. Wilson, Sep 26 2010

Keywords

• nonn

Comments

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

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

Examples

			11^10 has 11 divisors, so 11 is in the sequence.
225^7 has 225 divisors, so 225 is in the sequence.
		

Links

• David W. Wilson, Table of n, a(n) for n=1..10000

Crossrefs

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

A180935 gives the corresponding k.

Cf. A036878, A046315.

Programs

Mathematica

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 *)