A368925 The minimal exponent in the prime factorization of the powerful numbers.

0, 2, 3, 2, 4, 2, 3, 5, 2, 2, 6, 2, 4, 2, 2, 2, 3, 7, 2, 2, 2, 2, 3, 2, 5, 8, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 9, 2, 2, 4, 3, 2, 2, 6, 2, 2, 2, 3, 2, 2, 2, 2, 3, 10, 2, 2, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 11, 2, 7, 3, 2, 2, 2, 4

Offset: 1

Author

Amiram Eldar, Feb 15 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A051904, A368710.

Programs

Mathematica

s[n_] := If[n == 1, 0, Min @@ Last /@ FactorInteger[n]]; s /@ Select[Range[3000], # == 1 || Min[FactorInteger[#][[;;, 2]]] > 1 &]
(* or *)
f[n_] := Module[{e = Min[FactorInteger[n][[;; , 2]]]}, If[n == 1, 0, If[e > 1, e, Nothing]]]; Array[f, 3000]

PARI

lista(kmax) = {my(e); for(k = 1, kmax, if(k == 1, print1(0, ", "), e = vecmin(factor(k)[,2]); if(e > 1, print1(e, ", "))));}

Formula

a(n) = A051904(A001694(n)).

a(n) >= 2 for n >= 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.