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Numbers k such that there are exactly two biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.
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%I #8 Oct 24 2020 04:03:38

%S 14,31,67,72,82,93,98,110,132,140,156,172,189,192,223,240,257,281,285,

%T 322,347,368,379,407,410,414,426,441,455,468,472,481,488,514,515,517,

%U 524,525,537,551,555,574,579,602,613,664,680,693,702,703,737,743,749,755

%N Numbers k such that there are exactly two biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

%C Positions of 2's in A338326.

%C The asymptotic density of this sequence is 0.058757863... (Dehkordi, 1998).

%H Amiram Eldar, <a href="/A338389/b338389.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 14 is a term since there are exactly two biquadratefree powerful numbers, 200 = 2*3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and (14+1)^2 = 225.

%t bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[800], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 2 &]

%Y Cf. A336177, A338325, A338326, A338327, A338387, A338388, A338390, A338391, A338392.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 23 2020