A338326 - OEIS

%I #10 Oct 23 2020 03:28:25

%S 0,1,0,0,1,0,0,1,0,1,1,0,0,2,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,2,0,1,0,

%T 0,3,1,0,0,0,0,1,0,0,0,1,0,1,0,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0,2,0,

%U 0,1,0,2,1,0,0,0,0,1,0,0,0,2,0,0,0,0,1

%N The number of biquadratefree powerful numbers (A338325) between the consecutive squares n^2 and (n+1)^2.

%C Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that a(m) = k has a positive asymptotic density.

%H Amiram Eldar, <a href="/A338326/b338326.txt">Table of n, a(n) for n = 1..10000</a>

%H Massoud H. Dehkordi, <a href="https://hdl.handle.net/2134/12177">Asymptotic formulae for some arithmetic functions in number theory</a>, Ph.D. thesis, Loughborough University, 1998.

%e a(2) = 1 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 9.

%t bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; a[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; Array[a, 100]

%Y Cf. A119241, A337736, A338325.

%K nonn

%O 1,14

%A _Amiram Eldar_, Oct 22 2020