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%I #11 Oct 29 2020 04:54:24
%S 1,2,14,36,234,3510,211297,487425,20136429
%N a(n) is the least number k such that there are exactly n biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.
%C a(n) is the least k such that A338326(k) = n.
%C Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that A338326(m) = k has a positive asymptotic density. Therefore, this sequence is infinite.
%C a(9) > 10^10. - _Bert Dobbelaere_, Oct 29 2020
%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(0) = 1 since there are no biquadratefree powerful numbers between 1^2 = 1 and 2^2 = 4.
%e a(1) = 2 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 8.
%e a(2) = 14 since there are 2 biquadratefree powerful numbers, 200 = 2^3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and 15^2 = 225.
%t bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; f[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; mx = 5; s = Table[0, {mx}]; c = 0; n = 1; While[c < mx, i = f[n] + 1; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++]; s
%Y Cf. A119242, A337737, A338325, A338326.
%K nonn,more
%O 0,2
%A _Amiram Eldar_, Oct 22 2020
%E a(8) from _Bert Dobbelaere_, Oct 29 2020
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