A270337 Composite numbers equal to the number of divisors of one of their powers.

9, 25, 28, 40, 45, 49, 81, 121, 153, 169, 225, 289, 325, 343, 361, 441, 529, 625, 640, 841, 961, 976, 1089, 1225, 1369, 1521, 1681, 1849, 2133, 2197, 2209, 2401, 2541, 2601, 2809, 3025, 3249, 3481, 3721, 4225, 4489, 4753, 4761, 4851, 5041, 5329, 5929, 6241, 6348, 6561, 6859, 6889

Offset: 1

Author

Paolo P. Lava, Mar 15 2016

Keywords

• nonn
• easy

Comments

Prime numbers are not considered since every prime p satisfies p = d(p^(p-1)), where d() represents the number of divisors.

In general, p^k = d((p^k)^((p^k-1)/k)) for any prime p and for any power k such that (p^k-1)/k is an integer.

Examples

			9 = d(9^4); 28 = d(28^3); 153 = d(153^8); etc.
		

Links

• Paolo P. Lava, First 50 terms with their powers

Crossrefs

Cf. A000005, A073049, A270389.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n;
for n from 2 to q do if not isprime(n) then a:=tau(n); k:=0;
while a
				

Mathematica

nn = 2000; Select[Select[Range@ nn, CompositeQ], Function[k, (SelectFirst[k^Range[nn/2], DivisorSigma[0, #] == k &] /. n_ /; MissingQ@ n -> 0) > 0]] (* Michael De Vlieger, Mar 17 2016, Version 10.2 *)