cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-36 of 36 results.

A024839 Least m such that if r and s in {1/4, 1/8, 1/12, ..., 1/4n} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

Original entry on oeis.org

13, 33, 61, 97, 161, 221, 313, 393, 513, 613, 761, 881, 1057, 1249, 1405, 1625, 1861, 2049, 2313, 2593, 2813, 3121, 3445, 3697, 4049, 4417, 4801, 5101, 5513, 5941, 6385, 6729, 7201, 7689, 8193, 8581, 9113, 9661, 10225, 10657, 11249, 11857
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 12 2012

Links

Crossrefs

Cf. A001000, A024838.

Programs

  • Mathematica
    leastSeparatorS[seq_, s_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Map[leastSeparatorS[1/(4*Range[50]), #] &, Range[5]];
t[[2]] (* A024839 *)
(* Peter J. C. Moses, Aug 06 2012 *)

Extensions

Corrected by Clark Kimberling, Aug 12 2012

A024840 a(n) = least m such that if r and s in {1/1, 1/2, 1/3, ..., 1/n} satisfy r < s, then r < k/m < (k+2)/m < s for some integer k.

Original entry on oeis.org

7, 17, 31, 49, 71, 97, 127, 169, 209, 262, 311, 375, 433, 508, 575, 661, 737, 834, 919, 1027, 1141, 1241, 1366, 1497, 1611, 1753, 1901, 2029, 2188, 2353, 2495, 2671, 2853, 3009, 3202, 3401, 3571, 3781, 3997, 4219, 4409, 4642, 4881, 5126, 5335, 5591, 5853, 6121, 6349, 6628
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 08 2012

Examples

			Using the terminology introduced at A001000, the 3rd separator of the set {1/3, 1/2, 1} is a(3) = 17, since 1/3 < 6/17 < 8/17 < 1/2 < 8/17 < 10/17 < 1 and 17 is the least m for which 1/3, 1/2, 1 are thus separated using numbers k/m. - _Clark Kimberling_, Aug 08 2012

Links

Crossrefs

Cf. A001000.

Programs

  • Mathematica
    leastSeparatorS[seq_, s_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Map[leastSeparatorS[1/Range[50], #] &, Range[5]];
TableForm[t]
t[[3]] (* Peter J. C. Moses, Aug 08 2012 *)

A024848 a(n) = least m such that if r and s in {1/2, 1/4, 1/6, ..., 1/2n} satisfy r < s, then r < k/m < (k+4)/m < s for some integer k.

Original entry on oeis.org

19, 53, 103, 169, 251, 349, 463, 593, 739, 901, 1101, 1299, 1537, 1769, 2045, 2311, 2625, 2925, 3277, 3611, 4001, 4369, 4797, 5199, 5665, 6101, 6605, 7075, 7617, 8121, 8701, 9301, 9859, 10497, 11155, 11765, 12461, 13177, 13839, 14593, 15367, 16081, 16893, 17725
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 12 2012

Links

Crossrefs

Cf. A001000, A024847.

Programs

  • Mathematica
    leastSeparatorS[seq_, s_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Map[leastSeparatorS[1/(2*Range[50]), #] &, Range[5]];
TableForm[t]
t[[5]]  (* A024848 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A214921 Least m > 0 such that for every r and s in the set S = {{h*sqrt(2)} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

2, 3, 4, 5, 7, 7, 12, 12, 12, 12, 12, 15, 15, 17, 17, 17, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 33, 36, 36, 36, 41, 41, 41, 41, 41, 41, 41, 41, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70
Offset: 2

Views

Author

Clark Kimberling, Aug 12 2012

Keywords

Comments

a(n) is the least separator of S, as defined at A001000, which includes a guide to related sequences. - Clark Kimberling, Aug 12 2012

Examples

			Write the fractional parts of h*sqrt(2) for h=1,2,...,6, sorted, as f1, f2, f3, f4, f5, f6. Then f1 < 1/7 < f2 < 2/7 < f3 < 3/7 < f4 < 4/7 < f5 < 5/7 < f6, and 7 is the least m for which such a separation by fractions k/m occurs, so that a(6)=7.

Links

Crossrefs

Cf. A001000, A214964, A214965.

Programs

  • Mathematica
    leastSeparatorShort[seq_, s_] := Module[{n = 1},
While[Or @@ (n #1[[1]] <= s + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[seq, 2, 1], n++]; n];
Table[leastSeparatorShort[Sort[N[FractionalPart[Sqrt[2] Range[n]], 50]], 1], {n, 2, 100}]
(* Peter J. C. Moses, Aug 01 2012 *)

A024831 a(n) = least m such that if r and s in {F(h)/F(2*h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

2, 7, 10, 10, 15, 23, 37, 59, 95, 153, 247, 399, 645, 1043, 1687, 2729, 4415, 7143, 11557, 18699, 30255, 48953, 79207, 128159, 207365, 335523, 542887, 878409, 1421295, 2299703, 3720997, 6020699, 9741695, 15762393, 25504087, 41266479, 66770565, 108037043, 174807607, 282844649
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

Note that F(2*h)/F(h) = Lucas(h) for h > 0. - Editors.
For a guide to related sequences, see A001000. - Clark Kimberling, Aug 07 2012

Links

Crossrefs

Cf. A001000, A001622, A022112.

Programs

  • Mathematica
    leastSeparator[seq_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
2 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Table[N[Fibonacci[h]/Fibonacci[2 h]], {h, 1, 30}]
t1 = leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

Formula

From Philippe Deléham, Feb 06 2024: (Start)
a(n) = a(n-1) + a(n-2) - 1 for n >= 8.
a(n) = 2*a(n-1) - a(n-3) for n >= 9.
a(n) = 1 + A022112(n-3) for n >= 6.
a(n) = floor(((1 + sqrt(5))/2)*a(n-1)) for n >= 8.
G.f.: x^2*(x^6+3*x^5+2*x^4-8*x^3-4*x^2+3*x+2)/((x-1)*(x^2+x-1)).
(End)

Extensions

All the terms were corrected by Clark Kimberling, Aug 07 2012
More terms from Sean A. Irvine, Jul 25 2019

A024849 a(n) = least m such that if r and s in {|F(h+1)-tau*F(h)|: h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers) and tau = (1+sqrt(5))/2 (golden ratio).

Original entry on oeis.org

2, 4, 6, 9, 14, 23, 36, 59, 94, 153, 246, 399, 644, 1043, 1686, 2729, 4414, 7143, 11556, 18699
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 12 2012

Crossrefs

Cf. A000045, A001000, A001622.

Programs

  • Mathematica
    f[n_] := Fibonacci[n]; r = GoldenRatio;
leastSeparator[seq_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
2 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Flatten[Table[Abs[f[h + 1] - r*f[h]], {h, 1, 21}]];
leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)
Previous Showing 31-36 of 36 results.

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