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

1,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000005, A000312, A062319, A246655.

Sequence in context: A180040 A166274 A376595 * A039168 A059589 A153287

Adjacent sequences: A343729 A343730 A343731 * A343733 A343734 A343735

Keyword

nonn

Author

Jon E. Schoenfield, Jun 01 2021

Status

approved