A338327 a(n) is the least number k such that there are exactly n biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

1, 2, 14, 36, 234, 3510, 211297, 487425, 20136429

Offset: 0

Author

Amiram Eldar, Oct 22 2020

Keywords

• nonn
• more

Comments

a(n) is the least k such that A338326(k) = n.

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.

a(9) > 10^10. - Bert Dobbelaere, Oct 29 2020

Examples

			a(0) = 1 since there are no biquadratefree powerful numbers between 1^2 = 1 and 2^2 = 4.
a(1) = 2 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 8.
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.
		

Links

• Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Crossrefs

Cf. A119242, A337737, A338325, A338326.

Programs

Mathematica

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

Extensions

a(8) from Bert Dobbelaere, Oct 29 2020