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%I #16 Oct 29 2022 14:32:53
%S 2,3,4,6,7,8,9,10,15,22,26,30,31,36,42,46,58,66,70,78,82,102,106,121,
%T 127,130,138,166,178,190,210,222,226,238,255,262,282,310,330,346,358,
%U 366,382,418,430,438,441,442,462,466,478,498,502,511,546,562,570,586
%N Numbers k at which tau(k^k) is a prime power, where tau is the number-of-divisors function A000005.
%H Harvey P. Dale, <a href="/A343732/b343732.txt">Table of n, a(n) for n = 1..1000</a>
%e 9^9 = (3^2)^9 = 3^18 has 19 = 19^1 divisors, so 9 is a term.
%e 10^10 = 2^10 * 5^10 has 121 = 11^2 divisors, so 10 is a term.
%e 11^11 has 12 = 2^2 * 3^1 divisors, so 11 is not a term.
%t a={}; For[k=1,k<600,k++,If[PrimePowerQ[DivisorSigma[0,k^k]],AppendTo[a,k]]]; a (* _Stefano Spezia_, Jun 02 2021 *)
%t Select[Range[600],PrimePowerQ[DivisorSigma[0,#^#]]&] (* _Harvey P. Dale_, Oct 29 2022 *)
%o (PARI) isok(k) = isprimepower(numdiv(k^k)); \\ _Michel Marcus_, Jun 02 2021
%o (Python)
%o from functools import reduce
%o from operator import mul
%o from sympy import factorint
%o 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
%Y Cf. A000005, A000312, A062319, A246655.
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jun 01 2021
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