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A366196
The number of ways to express n^n in the form a^b for positive integers a and b.
2
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, 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, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 15, 4, 2, 12, 4
OFFSET
2,1
EXAMPLE
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.
MAPLE
a:= n-> numtheory[tau](igcd(map(i-> i[2], ifactors(n)[2])[])*n):
seq(a(n), n=2..100); # Alois P. Heinz, Oct 03 2023
MATHEMATICA
intPowCountPos[n_] := Module[{m, F, i, t},
m = n (GCD @@ FactorInteger[n][[All, 2]]);
t = 0;
While[Mod[m, 2] == 0,
t++;
m = m/2];
t = t + 1;
F = FactorInteger[m][[All, 2]];
If[m > 1,
For[i = 1, i <= Length[F], i++,
t = t (F[[i]] + 1)];
];
Return[t]]
PROG
(Python)
from math import gcd
from sympy import divisor_count, factorint
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
CROSSREFS
Sequence in context: A318312 A318474 A326306 * A278525 A357531 A318476
KEYWORD
nonn
AUTHOR
Andy Niedermaier, Oct 03 2023
STATUS
approved