A382906 The powerful part of the n-th biquadratefree number.

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1

Offset: 1

Author

Amiram Eldar, Apr 08 2025

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000000 terms)

Crossrefs

Cf. A013662, A046100, A057521.

Similar sequences: A382902, A382903, A382904, A382905.

Programs

Mathematica

f[p_, e_] := p^If[e > 1, e, 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] > 1, f[i, 2], 0)), ", ")));}

Formula

a(n) = A057521(A046100(n)).

a(n) = A046100(n)/A382905(n).

Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(4)^(3/2) * Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 2.21177275344948791706... .