A183162 Least integer k such that floor(k*sqrt(n+1)) > k*sqrt(n).

1, 3, 2, 1, 5, 3, 2, 3, 1, 7, 4, 3, 2, 3, 4, 1, 9, 5, 3, 5, 2, 3, 4, 5, 1, 11, 6, 4, 3, 5, 2, 5, 3, 4, 6, 1, 13, 7, 5, 4, 3, 7, 2, 5, 3, 4, 5, 7, 1, 15, 8, 5, 4, 3, 5, 7, 2, 5, 3, 7, 4, 6, 8, 1, 17, 9, 6, 5, 4, 3, 5, 7, 2, 5, 8, 3, 4, 5, 6, 9, 1, 19, 10, 7, 5, 4, 7, 3, 5, 9

Offset: 0

Author

Clark Kimberling, Dec 27 2010

Keywords

• nonn

Comments

a(n) is the least positive integer k such that one of the following holds:

(1) there is an integer J such that n*k^2 < J^2 < (n+1)*k^2; or

(2) there is an integer J such that (n+1)*k^2 = J^2.

Note that (1) is equivalent to the existence of a rational number H with denominator k such that n < H^2 < n+1.

Positions of 1: A005563.

Positions of 2: 2*A000217.

Positions of 2n+1: A000290.

Examples

			The results are easily read from an array of k*sqrt(n),
represented here by approximations:
1.00 1.41 1.73 2.00 2.24 2.45 2.65
2.00 2.83 3.46 4.00 4.47 4.90 5.29
3.00 4.24 5.20 6.00 6.71 7.35 7.94
4.00 5.66 6.93 8.00 8.94 9.80 10.58
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..1000
• Michael Weiss, On the Distribution of Rational Squares, arXiv:1510.07362 [math.NT], 2015.
• Michael Weiss, Where Are the Rational Squares?, The American Mathematical Monthly, Vol. 124, No. 3 (March 2017), pp. 255-259.

Crossrefs

Cf. A183163, A183164, A005563, A000217, A000290, A275817.

Programs

Mathematica

Table[k = 1; While[Floor[k Sqrt[n + 1]] <= k Sqrt@ n, k++]; k, {n, 120}] (* Michael De Vlieger, Aug 14 2016 *)

PARI

a(n) = my(k = 1); while(floor(k*sqrt(n+1)) <= k*sqrt(n), k++); k; \\ Michel Marcus, Oct 07 2017

Extensions

Added a(0)=1 and changed b-file by N. J. A. Sloane, Aug 16 2016