A374587 The maximum exponent in the prime factorization of the numbers that are not coprime to the maximum exponent in their prime factorization.

2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 3, 2, 6, 2, 3, 2, 4, 2, 2, 2, 2, 2, 3, 4, 2, 3, 2, 2, 2, 3, 2, 4, 2, 2, 2, 5, 4, 2, 3, 2, 4, 2, 2, 3, 6, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 4, 2, 2, 2, 8, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 3, 2, 6, 4, 2, 4, 2, 2, 2, 2

Offset: 1

Author

Amiram Eldar, Jul 12 2024

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A051903, A368715, A374586.

Programs

Mathematica

f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[!CoprimeQ[n, e], e, Nothing]]; Array[f, 350]

PARI

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

Formula

a(n) = A051903(A368715(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k * d(k) / Sum_{k>=2} d(k) = 2.74240523513766773312..., where d(k) = (1 - 1/f(k+1, k))/zeta(k+1) - (1 - 1/f(k, k))/zeta(k), and f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).