A383005 Exponent of the highest power of 2 dividing the n-th biquadratefree number.

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0

Offset: 1

Author

Amiram Eldar, Apr 12 2025

Keywords

• nonn
• easy

Comments

First differs from A254990 at n = 31.

Links

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

Crossrefs

Cf. A046100, A007814, A254990, A373550, A383004.

Programs

Mathematica

biqFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 4 &]; IntegerExponent[Select[Range[200], biqFreeQ], 2]

PARI

isbiqfree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 3, return (0))); 1; }
list(lim) = for(k = 1, lim, if(isbiqfree(k), print1(valuation(k, 2), ", ")));

Formula

a(n) = A007814(A046100(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 11/15.

In general, the asymptotic mean of the exponent of the highest power of 2 dividing the n-th k-free number (number that is not divisible by a k-th power other than 1), for k >= 2, is 1 - k/(2^k-1).