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A374586
The maximum exponent in the prime factorization of the numbers that are divisible by the maximum exponent in their prime factorization.
2
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
1,3
LINKS
FORMULA
a(n) = A051903(A336064(n)).
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.
MATHEMATICA
f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e > 0 && Divisible[n, e], e, Nothing]]; Array[f, 150]
PROG
(PARI) lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(!(k % e), print1(e, ", "))); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jul 12 2024
STATUS
approved