A374586 The maximum exponent in the prime factorization of the numbers that are divisible by the maximum exponent in their prime factorization.
1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e > 0 && Divisible[n, e], e, Nothing]]; Array[f, 150] -
PARI
lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(!(k % e), print1(e, ", ")));}
Formula
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) / Sum_{k>=1} d(k) = 1.46254458894503050564..., where d(k) = (1/zeta(k+1)) * Product_{p prime, p|k} ((p^(k-e(p,k)+1) - 1)/(p^(k+1) - 1)) - (1/zeta(k)) * Product_{p prime, p|k} ((p^(k-e(p,k)) - 1)/(p^k - 1)) for k >= 2, and d(1) = 1/zeta(2), and e(p,k) is the exponent of the largest of power of p that divides k.