A336176 Numbers k such that there is a single powerful number between k^2 and (k+1)^2.

2, 8, 10, 15, 16, 18, 19, 20, 28, 29, 37, 39, 41, 42, 45, 48, 50, 51, 52, 53, 56, 57, 59, 63, 65, 74, 76, 77, 78, 79, 83, 84, 87, 89, 90, 92, 94, 100, 101, 102, 107, 113, 114, 115, 116, 117, 118, 119, 121, 122, 126, 127, 130, 134, 138, 141, 144, 146, 147, 148

Offset: 1

Author

Amiram Eldar, Jul 10 2020

Keywords

• nonn

Comments

Positions of 1's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density 0.3955... A more accurate calculation using his formula gives 0.3955652153962362...

1 is the most common value of A119241.

Examples

			2 is a term since there is a single powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.
		

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, A336177, A336178.

Programs

Mathematica

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[150], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 1 &]