A382889 The largest square dividing the n-th cubefree number.

1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 25, 1, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 49, 9, 100

Offset

1,4

Comments

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

All the terms are squares of squarefree numbers (A062503).

Links

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

Formula

a(n) = A008833(A004709(n)).

a(n) = A057521(A004709(n)).

a(n) = A382890(n)^2.

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

a(n) = (A004709(n)/A382888(n))^2.

a(A371188(n)) = 1.

Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .

Mathematica

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

Prog

(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, 2)), ", "))); }

Crossrefs

Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).

Similar sequences: A382888, A382890, A382891.

Sequence in context: A057521 A393292 A387716 * A084885 A382219 A360969

Adjacent sequences: A382886 A382887 A382888 * A382890 A382891 A382892

Keyword

nonn,easy

Author

Amiram Eldar, Apr 07 2025

Status

approved