A336178 Numbers k such that there are exactly three powerful numbers between k^2 and (k+1)^2.

31, 36, 67, 93, 132, 140, 145, 161, 166, 189, 192, 220, 223, 265, 280, 290, 296, 311, 316, 322, 364, 384, 407, 468, 537, 576, 592, 602, 623, 639, 644, 656, 659, 661, 670, 690, 722, 769, 771, 793, 828, 883, 888, 890, 896, 950, 961, 981, 984, 987, 992, 995, 1018

Offset

1,1

Comments

Positions of 3's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density = 0.0770... A more accurate calculation using his formula gives 0.0770742722233...

References

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Links

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

P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Example

31 is a term since there are exactly three powerful numbers, 968 = 2^3 * 11^2, 972 = 2^2 * 3^5 and 1000 = 2^3 * 5^3 between 31^2 = 961 and (31+1)^2 = 1024.

Mathematica

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[1000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 3 &]

Crossrefs

Cf. A001694, A119241, A119242, A336175, A336176, A336177.

Sequence in context: A300093 A109293 A053684 * A044973 A230035 A051267

Adjacent sequences: A336175 A336176 A336177 * A336179 A336180 A336181

Keyword

nonn

Author

Amiram Eldar, Jul 10 2020

Status

approved