A382905 The powerfree part of the n-th biquadratefree number.

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 65, 66, 67, 17, 69, 70, 71, 1, 73, 74, 3, 19

Offset: 1

Author

Amiram Eldar, Apr 08 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A013662, A046100, A055231.

Similar sequences: A382902, A382903, A382904, A382906.

Programs

Mathematica

f[p_, e_] := p^If[e < 2, 1, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 4 &], 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]) < 4, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] < 2, f[i, 2], 0)), ", ")));}

Formula

a(n) = A055231(A046100(n)).

a(n) = A046100(n)/A382906(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^5 + 1/p^6 - 1/p^7) = 0.75836309584506672244... .