A382890 The square root of the largest square dividing the n-th cubefree number.

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 7, 3, 10, 1, 1

Offset: 1

Author

Amiram Eldar, Apr 07 2025

Keywords

• nonn
• easy

Comments

The product of the non-unitary prime divisors of the n-th cubefree number.

Also, the square root of the powerful part of the n-th cubefree number.

All the terms are squarefree.

Links

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

Crossrefs

Cf. A000188, A004709, A005117, A057521, A371188 (positions of 1's).

Similar sequences: A382888, A382889, A382891.

Programs

Mathematica

f[p_, e_] := p^If[e == 1, 0, 1]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

PARI

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 1)), ", ")));}

Formula

a(n) = A000188(A004709(n)).

a(n) = sqrt(A382889(n)).

a(n) = A004709(n)/A382888(n).

a(n) = sqrt(A004709(n)/A382891(n)).

a(A371188(n)) = 1.