A366196 - OEIS

%I #12 Oct 04 2023 19:04:02

%S 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 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 6,4,8,2,12,2,4,6,6,4,8,2,10,15,4,2,12,4

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

%e 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 a:= n-> numtheory[tau](igcd(map(i-> i[2], ifactors(n)[2])[])*n):

%p seq(a(n), n=2..100);  # _Alois P. Heinz_, Oct 03 2023

%t intPowCountPos[n_] := Module[{m, F, i, t},

%t   m = n (GCD @@ FactorInteger[n][[All, 2]]);

%t   t = 0;

%t   While[Mod[m, 2] == 0,

%t    t++;

%t    m = m/2];

%t   t = t + 1;

%t   F = FactorInteger[m][[All, 2]];

%t   If[m > 1,

%t    For[i = 1, i <= Length[F], i++,

%t      t = t (F[[i]] + 1)];

%t    ];

%t   Return[t]]

%o (Python)

%o from math import gcd

%o from sympy import divisor_count, factorint

%o 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 Cf. A000312, A366161.

%K nonn

%O 2,1

%A _Andy Niedermaier_, Oct 03 2023