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A366196 The number of ways to express n^n in the form a^b for positive integers a and b.

%I A366196 #12 Oct 04 2023 19:04:02
%S A366196 2,2,4,2,4,2,8,6,4,2,6,2,4,4,7,2,6,2,6,4,4,2,8,6,4,5,6,2,8,2,12,4,4,4,
%T A366196 12,2,4,4,8,2,8,2,6,6,4,2,10,6,6,4,6,2,8,4,8,4,4,2,12,2,4,6,16,4,8,2,
%U A366196 6,4,8,2,12,2,4,6,6,4,8,2,10,15,4,2,12,4
%N A366196 The number of ways to express n^n in the form a^b for positive integers a and b.
%e A366196 a(27) = 5, as "27^27 = a^b" has 5 positive integer solutions: 3^81, 27^27, 19683^9, 7625597484987^3, and (3^81)^1.
%p A366196 a:= n-> numtheory[tau](igcd(map(i-> i[2], ifactors(n)[2])[])*n):
%p A366196 seq(a(n), n=2..100);  # _Alois P. Heinz_, Oct 03 2023
%t A366196 intPowCountPos[n_] := Module[{m, F, i, t},
%t A366196   m = n (GCD @@ FactorInteger[n][[All, 2]]);
%t A366196   t = 0;
%t A366196   While[Mod[m, 2] == 0,
%t A366196    t++;
%t A366196    m = m/2];
%t A366196   t = t + 1;
%t A366196   F = FactorInteger[m][[All, 2]];
%t A366196   If[m > 1,
%t A366196    For[i = 1, i <= Length[F], i++,
%t A366196      t = t (F[[i]] + 1)];
%t A366196    ];
%t A366196   Return[t]]
%o A366196 (Python)
%o A366196 from math import gcd
%o A366196 from sympy import divisor_count, factorint
%o A366196 def A366196(n): return divisor_count((m:=n*gcd(*factorint(n).values()))>>(t:=(m-1&~m).bit_length()))*(t+1) # _Chai Wah Wu_, Oct 04 2023
%Y A366196 Cf. A000312, A366161.
%K A366196 nonn
%O A366196 2,1
%A A366196 _Andy Niedermaier_, Oct 03 2023