A183163 -id:A183163 - cp's OEIS frontend

A183162 Least integer k such that floor(ksqrt(n+1)) > ksqrt(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

Clark Kimberling, Dec 27 2010

nonn

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.

			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

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.

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

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

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

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

A183164 Least integer k such that karctan(n) and karctan(n+1) are separated by an integer.

Original entry on oeis.org

1, 5, 4, 3, 5, 7, 9, 11, 13, 19, 27, 41, 85, 2, 61, 43, 33, 29, 25, 23, 21, 19, 36, 17, 32, 15, 43, 28, 41, 13, 76, 50, 37, 24, 35, 46, 57, 101, 11, 97, 64, 53, 42, 31, 51, 71, 111, 20, 69, 49, 78, 29, 67, 105, 38, 47, 103, 56, 74, 83, 101, 128, 182, 299, 9

Offset: 1

Clark Kimberling, Dec 27 2010

nonn

a(n) is the least positive integer for which there is a rational number H with denominator k for which n < tan(H) < n+1.

Cf. A183162, A183163, A183201.

Table[k=1; While[Floor[k*ArcTan[n+1]]<=k*ArcTan[n], k++];k,{n,100}]

A183200 Least integer k such that floor(kf(n+1))>kf(n), where f(n)=log_2 of n.

Original entry on oeis.org

1, 2, 1, 4, 2, 3, 1, 6, 4, 3, 2, 3, 4, 6, 1, 12, 6, 5, 4, 3, 5, 2, 7, 5, 3, 4, 5, 6, 8, 11, 1, 23, 12, 8, 6, 5, 9, 4, 7, 3, 8, 5, 7, 13, 2, 11, 7, 5, 8, 3, 10, 7, 4, 9, 5, 11, 6, 8, 9, 11, 15, 22, 1, 45, 23, 16, 12, 10, 8, 7, 6, 11, 5, 9, 13, 4, 11, 7, 13, 3, 14, 8, 13, 5, 12, 7, 9, 13, 21, 2, 23, 13, 11, 9, 7, 17, 5, 8, 11, 17

Offset: 1

Clark Kimberling, Dec 29 2010

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A183163.

  Table[k=1; While[Floor[k*Log[2,n+1]]<=k*Log[2,n], k++]; k, {n,100}]

A227634 Least splitter of log(n) and log(n+1).

Original entry on oeis.org

1, 1, 3, 2, 3, 5, 1, 6, 4, 3, 5, 2, 5, 3, 4, 5, 6, 10, 18, 1, 11, 8, 6, 5, 4, 7, 10, 3, 5, 7, 9, 15, 2, 11, 7, 5, 8, 14, 3, 10, 7, 4, 9, 5, 11, 6, 7, 8, 10, 12, 15, 21, 34, 1, 40, 24, 17, 13, 11, 10, 8, 7, 13, 6, 11, 5, 14, 9, 17, 4, 11, 7, 10, 13, 22, 3, 17

Offset: 1

Clark Kimberling, Jul 18 2013

Essentially the same as A183163. - R. J. Mathar, Jul 27 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

			The splitting rationals of consecutive numbers log(1), log(2), ... are 0, 1, 4/3, 3/2, 5/3, 9/5, 2, 13/6, 9/4, 7/3, 12/5, 5/2, 13/5; the denominators form A227634, and the numerators, A227684.  Chain:
log(1) <= 0 < log(2) < 1 < log(3) < 4/3 < log(4) < 3/2 < log(5) < 5/3 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631.

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[Log[n], Log[n + 1]], {n, 1, 120}] (*fractions*)
Denominator[t] (* A227634 *)
Numerator[t]   (* A227684 *)

cp's OEIS Frontend

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

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A183164 Least integer k such that karctan(n) and karctan(n+1) are separated by an integer.

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A183200 Least integer k such that floor(kf(n+1))>kf(n), where f(n)=log_2 of n.

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A227634 Least splitter of log(n) and log(n+1).

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cp's OEIS Frontend

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

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A183164 Least integer k such that k*arctan(n) and k*arctan(n+1) are separated by an integer.

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A183200 Least integer k such that floor(k*f(n+1))>k*f(n), where f(n)=log_2 of n.

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A227634 Least splitter of log(n) and log(n+1).

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

A183164 Least integer k such that karctan(n) and karctan(n+1) are separated by an integer.

A183200 Least integer k such that floor(kf(n+1))>kf(n), where f(n)=log_2 of n.