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

1,2

Links

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

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... .

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]

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] < 2, f[i, 2], 0)), ", "))); }

Crossrefs

Cf. A013662, A046100, A055231.

Similar sequences: A382902, A382903, A382904, A382906.

Sequence in context: A304339 A379457 A160400 * A325978 A326049 A380085

Adjacent sequences: A382902 A382903 A382904 * A382906 A382907 A382908

Keyword

nonn,easy

Author

Amiram Eldar, Apr 08 2025

Status

approved