A183162 -id:A183162 - cp's OEIS frontend

A183163 Least integer k such that floor(klog(n+1))>klog(n).

2, 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, Dec 27 2010

nonn

Equivalently, a(n) is the least integer k for which there is an integer J such that n^k < e^J < (n+1)^k; or, equivalently, such that there is a rational number H with denominator k for which log(n) < H < log(n+1).

Cf. A183162.

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

A183163 = lambda n: next(k for k in IntegerRange(1, infinity) if floor(k*log(n+1)) > k*log(n)) # D. S. McNeil, Dec 28 2010

A275817 Least positive integer s such that an integer square k^2 lies between s^2n and s^2(n+1), with s^2n < k^2 < s^2(n+1).

Original entry on oeis.org

2, 3, 2, 4, 5, 3, 2, 3, 6, 7, 4, 3, 2, 3, 4, 8, 9, 5, 3, 5, 2, 3, 4, 5, 10, 11, 6, 4, 3, 5, 2, 5, 3, 4, 6, 12, 13, 7, 5, 4, 3, 7, 2, 5, 3, 4, 5, 7, 14, 15, 8, 5, 4, 3, 5, 7, 2, 5, 3, 7, 4, 6, 8, 16, 17, 9, 6, 5, 4, 3, 5, 7, 2, 5, 8, 3, 4, 5, 6, 9, 18, 19, 10, 7, 5, 4, 7, 3

Offset: 0

Hugo Pfoertner, Aug 09 2016

nonn

The corresponding values of k are provided in A275818.

			a(0)=2, because 2^2*0 < 1^2 < 2^2*(0+1).

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. A183162, A275818.

Table[s = 1; While[Count[Range[n s^2 + 1, (n + 1) s^2 - 1], k_ /; IntegerQ@ Sqrt@ k] == 0, s++]; s, {n, 0, 120}] (* Michael De Vlieger, Aug 14 2016 *)

If n = k^2 - 1 and k > 0, then a(n) = 2*k, A183162(n) = 1; otherwise a(n) = A183162(n).

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}]

A183197 Least integer k such that Floor(k(n+1)^(1/3))>kn^(1/3).

Original entry on oeis.org

4, 3, 2, 3, 4, 6, 1, 13, 7, 5, 4, 3, 5, 7, 2, 9, 5, 3, 7, 4, 5, 6, 7, 9, 14, 1, 28, 14, 10, 8, 6, 5, 9, 4, 7, 13, 3, 8, 5, 7, 11, 2, 17, 9, 7, 5, 8, 11, 3, 10, 7, 4, 9, 5, 11, 6, 7, 8, 10, 12, 16, 24, 1, 49, 25, 17, 13, 10, 9, 8, 7, 6, 11, 5, 9, 4, 15, 7, 10, 16, 3, 14, 8, 13, 5, 7, 9, 11, 15, 27, 2, 17, 11, 9, 7, 12, 5, 8, 11, 14

Offset: 1

Clark Kimberling, Dec 29 2010

nonn

Cf. A183162

 Table[k=1; While[Floor[k*(n+1)^(1/3)]<=k*n^(1/3), k++]; k,{n,100}]

A183199 Least integer k such that Floor(kf(n+1))>kf(n), where f(n)=(n^2)/(1+n^2).

Original entry on oeis.org

3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502

Offset: 1

Clark Kimberling, Dec 29 2010

nonn

Appears to be essentially the same as A102305, A059100 and A010000. - R. J. Mathar, Jun 07 2011

Cf. A183162.

Table[k=1; While[Floor[k*((n+1)^2)/(1+(n+1)^2)]<=k*(n^2)/(1+(n^2)), k++]; k, {n,100}]

A183201 Least integer k such that Floor(kf(n+1))>kf(n), where f(n)=arccosh(n).

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 3, 5, 9, 1, 6, 4, 7, 3, 7, 2, 7, 5, 3, 7, 4, 5, 6, 8, 12, 21, 1, 17, 11, 8, 7, 6, 5, 9, 4, 7, 13, 3, 8, 5, 7, 9, 11, 23, 2, 13, 9, 7, 5, 13, 8, 17, 3, 10, 7, 11, 4, 13, 9, 5, 11, 6, 13, 7, 8, 9, 10, 12, 14, 18, 23, 34, 61, 1, 42, 28, 21, 16, 14, 12, 11, 9, 17, 8, 7, 13, 6, 17, 11, 5, 14, 9, 13, 21, 4, 15, 11, 7, 17, 10

Offset: 1

Clark Kimberling, Dec 29 2010

nonn

Equivalently, k is the least positive integer such that there is a rational number H with denominator k for which n < cosh(H) < n+1.

Cf. A183162, A183164.

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

cp's OEIS Frontend

A183163 Least integer k such that floor(k*log(n+1))>k*log(n).

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A275817 Least positive integer s such that an integer square k^2 lies between s^2*n and s^2*(n+1), with s^2*n < k^2 < s^2*(n+1).

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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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A183197 Least integer k such that Floor(k*(n+1)^(1/3))>k*n^(1/3).

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A183199 Least integer k such that Floor(k*f(n+1))>k*f(n), where f(n)=(n^2)/(1+n^2).

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A183201 Least integer k such that Floor(k*f(n+1))>k*f(n), where f(n)=arccosh(n).

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A183163 Least integer k such that floor(klog(n+1))>klog(n).

A275817 Least positive integer s such that an integer square k^2 lies between s^2n and s^2(n+1), with s^2n < k^2 < s^2(n+1).

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

A183197 Least integer k such that Floor(k(n+1)^(1/3))>kn^(1/3).

A183199 Least integer k such that Floor(kf(n+1))>kf(n), where f(n)=(n^2)/(1+n^2).

A183201 Least integer k such that Floor(kf(n+1))>kf(n), where f(n)=arccosh(n).