A183162 - OEIS

%I #28 Oct 07 2017 03:37:45

%S 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,

%T 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,

%U 5,4,3,5,7,2,5,8,3,4,5,6,9,1,19,10,7,5,4,7,3,5,9

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

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

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

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

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

%C Positions of 1:  A005563.

%C Positions of 2: 2*A000217.

%C Positions of 2n+1: A000290.

%H Hugo Pfoertner, <a href="/A183162/b183162.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Weiss, <a href="http://arxiv.org/abs/1510.07362">On the Distribution of Rational Squares</a>, arXiv:1510.07362 [math.NT], 2015.

%H Michael Weiss, <a href="https://dx.doi.org/10.4169/amer.math.monthly.124.3.255">Where Are the Rational Squares?</a>, The American Mathematical Monthly, Vol. 124, No. 3 (March 2017), pp. 255-259.

%e The results are easily read from an array of k*sqrt(n),

%e represented here by approximations:

%e 1.00 1.41 1.73 2.00 2.24 2.45 2.65

%e 2.00 2.83 3.46 4.00 4.47 4.90 5.29

%e 3.00 4.24 5.20 6.00 6.71 7.35 7.94

%e 4.00 5.66 6.93 8.00 8.94 9.80 10.58

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

%o (PARI) a(n) = my(k = 1); while(floor(k*sqrt(n+1)) <= k*sqrt(n), k++); k; \\ _Michel Marcus_, Oct 07 2017

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

%K nonn

%O 0,2

%A _Clark Kimberling_, Dec 27 2010

%E Added a(0)=1 and changed b-file by _N. J. A. Sloane_, Aug 16 2016