A369429 Square root of the largest square dividing the n-th powerful number.

1, 2, 2, 3, 4, 5, 3, 4, 6, 7, 8, 6, 9, 10, 6, 11, 5, 8, 12, 13, 14, 10, 6, 15, 9, 16, 12, 17, 18, 7, 19, 14, 20, 12, 21, 22, 10, 16, 23, 24, 25, 18, 15, 26, 27, 28, 20, 29, 12, 30, 31, 22, 18, 10, 32, 33, 15, 24, 34, 35, 36, 21, 11, 26, 37, 14, 38, 39, 28, 40

Offset: 1

Author

Amiram Eldar, Jan 23 2024

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A000188, A001694, A087050, A369430.

Cf. A002117, A078434.

Programs

Mathematica

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[2000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]
(* or *)
f[p_, e_] := p^Floor[e/2]; f[p_, 1] := 0; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 2000], # > 0 &]

PARI

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, f[i,1]^(f[i,2]\2)));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

Formula

a(n) = A000188(A001694(n)).

a(n) > 1 for n >= 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (15/(2*Pi^2))*(zeta(3)/zeta(3/2))^2 = 0.160894210785... .