A336177 Numbers k such that there are exactly two powerful numbers between k^2 and (k+1)^2.

5, 11, 14, 22, 25, 33, 44, 46, 55, 58, 62, 70, 72, 73, 82, 88, 96, 98, 103, 104, 109, 110, 111, 124, 129, 135, 155, 156, 158, 164, 172, 176, 178, 181, 187, 197, 203, 206, 207, 209, 212, 218, 240, 243, 248, 249, 254, 257, 259, 268, 277, 279, 281, 285, 288, 291

Offset: 1

Author

Amiram Eldar, Jul 10 2020

Keywords

• nonn

Comments

Positions of 2's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density 0.2312... A more accurate calculation using his formula gives 0.231299167354828...

Examples

			5 is a term since there are exactly two powerful numbers, 27 = 3^3 and 32 = 2^5 between 5^2 = 25 and (5+1)^2 = 36.
		

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.

Crossrefs

Cf. A001694, A119241, A119242, A336175, A336176, A336178.

Programs

Mathematica

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[300], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 2 &]