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

1,1

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.

Links

Table of n, a(n) for n=1..52.

Paolo P. Lava, First 50 terms with their powers

Example

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

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<n do k:=k+1; a:=tau(n^k); od; if n=a then print(n); fi; fi; od; end: P(10^6);

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 *)

Crossrefs

Cf. A000005, A073049, A270389.

Sequence in context: A321874 A020252 A076486 * A339842 A068529 A096059

Adjacent sequences: A270334 A270335 A270336 * A270338 A270339 A270340

Keyword

nonn,easy

Author

Paolo P. Lava, Mar 15 2016

Status

approved