A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jul 22 2025

Keywords

• nonn
• easy

Comments

First differs from A375428 at n = 64.

Differs from A368105 at n = 1, 36, 64, 72, 100, ... .

Except for a(1), all the terms are by definition squarefree numbers.

Links

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

Crossrefs

Cf. A005117, A051903, A070321, A365683, A368105, A375428, A378085.

Programs

Mathematica

a[n_] := Module[{k = Max[FactorInteger[n][[;; , 2]]]}, While[! SquareFreeQ[k], k--]; k]; a[1] = 0; Array[a, 100]

PARI

a(n) = if(n == 1, 0, my(k = vecmax(factor(n)[,2])); while(!issquarefree(k), k--); k);

Formula

a(n) = A051903(A365683(n)).

a(n) = A070321(A051903(n)) for n >= 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} A378085(k-1)*(1-1/zeta(k)) = 1.66055078443790141429... .