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 A183162 Least integer k such that floor(k*sqrt(n+1)) > k*sqrt(n). 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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. 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. EXAMPLE 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 MATHEMATICA Table[k = 1; While[Floor[k Sqrt[n + 1]] <= k Sqrt@ n, k++]; k, {n, 120}] (* Michael De Vlieger, Aug 14 2016 *) PROG (PARI) a(n) = my(k = 1); while(floor(k*sqrt(n+1)) <= k*sqrt(n), k++); k; \\ Michel Marcus, Oct 07 2017 CROSSREFS Cf. A183163, A183164, A005563, A000217, A000290, A275817. Sequence in context: A278817 A171746 A113977 * A019587 A102427 A080883 Adjacent sequences:  A183159 A183160 A183161 * A183163 A183164 A183165 KEYWORD nonn AUTHOR Clark Kimberling, Dec 27 2010 EXTENSIONS Added a(0)=1 and changed b-file by N. J. A. Sloane, Aug 16 2016 STATUS approved

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