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

1,4

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000000 terms)

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... .

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]

Prog

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

Crossrefs

Cf. A013662, A046100, A057521.

Similar sequences: A382902, A382903, A382904, A382905.

Sequence in context: A088440 A300253 A212173 * A366993 A274006 A203025

Adjacent sequences: A382903 A382904 A382905 * A382907 A382908 A382909

Keyword

nonn,easy

Author

Amiram Eldar, Apr 08 2025

Status

approved