A392278 The maximum exponent in the prime factorization of the smallest multiple of n that is an exponentially squarefree number.

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

Offset

1,4

Comments

a(n) is squarefree by definition for all n >= 2.

Links

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

Formula

a(n) = A051903(A365684(n)).

a(n) = A067535(A051903(n)) for n >= 2.

a(n) >= A386468(n), with equality if and only if n is an exponentially squarefree number (A209061).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/zeta(2) + Sum_{k>=2} A067535(k)*(1/zeta(k+1)-1/zeta(k)) = 1 + Sum_{k>=2} A378087(k-1)*(1-1/zeta(k)) = 1.75091716381406194674... .

Mathematica

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

Prog

(PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k; };
a(n) = if(n == 1, 0, s(vecmax(factor(n)[, 2])));

Crossrefs

Cf. A005117, A051903, A067535, A209061, A365684, A378087, A386468, A391899.

Sequence in context: A331178 A227350 A107345 * A358780 A336736 A381455

Adjacent sequences: A392275 A392276 A392277 * A392279 A392280 A392281

Keyword

nonn,easy

Author

Amiram Eldar, Jan 06 2026

Status

approved