A365685 a(n) is the smallest number k such that k*n is an exponentially squarefree number (A209061).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1

Offset

1,16

Links

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

Formula

Multiplicative with a(p^e) = p^A081221(e).

a(n) = A365684(n)/n.

a(n) >= 1, 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) = Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^f(k-1))/p^k) = 1.06562841319..., where f(k) = A081221(k) and f(0) = 0.

Mathematica

f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k - e; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

Crossrefs

Cf. A081221, A209061, A365684.

Sequence in context: A054977 A272901 A078315 * A346418 A288120 A156264

Adjacent sequences: A365682 A365683 A365684 * A365686 A365687 A365688

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 15 2023

Status

approved