A343732 Numbers k at which tau(k^k) is a prime power, where tau is the number-of-divisors function A000005.

2, 3, 4, 6, 7, 8, 9, 10, 15, 22, 26, 30, 31, 36, 42, 46, 58, 66, 70, 78, 82, 102, 106, 121, 127, 130, 138, 166, 178, 190, 210, 222, 226, 238, 255, 262, 282, 310, 330, 346, 358, 366, 382, 418, 430, 438, 441, 442, 462, 466, 478, 498, 502, 511, 546, 562, 570, 586

Offset: 1

Author

Jon E. Schoenfield, Jun 01 2021

Keywords

• nonn

Examples

			9^9 = (3^2)^9 = 3^18 has 19 = 19^1 divisors, so 9 is a term.
10^10 = 2^10 * 5^10 has 121 = 11^2 divisors, so 10 is a term.
11^11 has 12 = 2^2 * 3^1 divisors, so 11 is not a term.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A000312, A062319, A246655.

Programs

Mathematica

a={}; For[k=1,k<600,k++,If[PrimePowerQ[DivisorSigma[0,k^k]],AppendTo[a,k]]]; a (* Stefano Spezia, Jun 02 2021 *)
Select[Range[600],PrimePowerQ[DivisorSigma[0,#^#]]&] (* Harvey P. Dale, Oct 29 2022 *)

PARI

isok(k) = isprimepower(numdiv(k^k)); \\ Michel Marcus, Jun 02 2021

Python

from functools import reduce
from operator import mul
from sympy import factorint
A343732_list = [n for n in range(2,10**3) if len(factorint(reduce(mul,(n*d+1 for d in factorint(n).values())))) == 1] # Chai Wah Wu, Jun 03 2021