A001000 -id:A001000 - cp's OEIS frontend

A214964 Least m > 0 such that for every r and s in the set S = {{h*(1+sqrt(5))/2} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k; m is the least separator of S as defined at A001000.

2, 3, 4, 5, 6, 8, 8, 10, 10, 13, 13, 13, 16, 16, 16, 21, 21, 21, 21, 21, 28, 30, 30, 30, 34, 34, 34, 34, 34, 34, 34, 34, 34, 43, 45, 50, 50, 50, 50, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 72, 73, 73, 73, 81, 81, 81, 81, 81, 81, 89, 89, 89, 89

Offset: 2

Clark Kimberling, Aug 12 2012

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..300

Cf. A001000, A214961, A214965.

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[GoldenRatio*Range[n]], 50]], 1], {n, 2, 100}]
(* Peter J. C. Moses, Aug 01 2012 *)

A214965 Least m > 0 such that for every r and s in the set S = {{h*e} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k; m is the least separator of S as defined at A001000.

Original entry on oeis.org

2, 3, 4, 6, 6, 7, 11, 11, 11, 18, 18, 18, 18, 25, 25, 25, 25, 25, 25, 25, 25, 25, 32, 32, 32, 32, 32, 32, 32, 32, 32, 35, 35, 35, 39, 39, 39, 39, 55, 61, 61, 66, 68, 69, 69, 69, 70, 70, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71

Offset: 2

Clark Kimberling, Aug 12 2012

nonn

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

			Write the sorted fractional parts {h*e}, for h=1..5, as f1,f2,f3,f4,f5.  Then f1 < 2/6 < f2 < 3/6 < f3 < 4/6 < f5 < 5/6 < f6, and no such separation occurs using fractions k/m having m < 6; so a(5) = 6.

Clark Kimberling, Table of n, a(n) for n = 2..300

Cf. A001000, A214961, A214964.

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[E*Range[n]], 50]], 1], {n, 2, 100}]
(* Peter J. C. Moses, Aug 01 2012 *)

A024838 Least m such that if r and s in {1/3, 1/6, 1/9, ..., 1/3n} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

Original entry on oeis.org

10, 25, 46, 73, 121, 166, 235, 295, 385, 460, 571, 661, 793, 937, 1054, 1219, 1396, 1537, 1735, 1945, 2110, 2341, 2584, 2773, 3037, 3313, 3601, 3826, 4135, 4456, 4789, 5047, 5401, 5767, 6145, 6436, 6835, 7246, 7669, 7993, 8437, 8893, 9361, 9841, 10210, 10711, 11224

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024836, A024837.

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/(3*Range[50]), #] &, Range[5]];
t[[2]] (* A024838 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A024836 a(n) = least m such that if r and s in {1/1, 1/4, 1/7, ..., 1/(3n-2)} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

Original entry on oeis.org

3, 13, 29, 51, 92, 131, 193, 248, 331, 401, 505, 590, 715, 852, 963, 1121, 1291, 1427, 1618, 1821, 1981, 2205, 2441, 2625, 2882, 3151, 3361, 3651, 3953, 4267, 4511, 4846, 5193, 5552, 5829, 6209, 6601, 7005, 7315, 7740, 8177, 8626, 9087, 9441, 9923, 10417, 10923, 11441

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024837, A024838.

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/(3*Range[50]-2), #] &, Range[5]];
t[[2]] (* A024836 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A024837 a(n) = least m such that if r and s in {1/2, 1/5, 1/8, ..., 1/(3n-1)} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

Original entry on oeis.org

7, 21, 41, 67, 100, 155, 205, 281, 346, 443, 523, 641, 737, 876, 1027, 1149, 1321, 1505, 1651, 1856, 2073, 2243, 2481, 2731, 2993, 3197, 3480, 3775, 4082, 4321, 4649, 4989, 5341, 5613, 5986, 6371, 6768, 7073, 7491, 7921, 8363, 8702, 9165, 9640, 10127, 10626, 11009

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024836, A024838.

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/(3*Range[50]-1), #] &, Range[5]];
t[[2]] (* A024837 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A024847 a(n) = least m such that if r and s in {1/1, 1/3, 1/5, ..., 1/(2n-1)} satisfy r < s, then r < k/m < (k+4)/m < s for some integer k.

Original entry on oeis.org

8, 34, 76, 134, 208, 298, 404, 526, 664, 818, 1009, 1198, 1427, 1651, 1918, 2176, 2481, 2773, 3116, 3442, 3823, 4183, 4602, 4996, 5453, 5881, 6376, 6838, 7371, 7867, 8438, 8969, 9578, 10207, 10791, 11458, 12145, 12781, 13506, 14251, 14939, 15722, 16525, 17265

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024848.

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]-1), #] &, Range[5]];
t[[5]] (* A024847 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A024832 Least m such that if r and s in {Pi/2 - atn(h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

2, 3, 7, 10, 17, 21, 31, 43, 50, 65, 82, 91, 111, 133, 157, 170, 197, 226, 257, 273, 307, 343, 381, 421, 442, 485, 530, 577, 626, 651, 703, 757, 813, 871, 931, 962, 1025, 1090, 1157, 1226, 1297, 1333, 1407, 1483, 1561, 1641, 1723, 1807, 1850, 1937, 2026, 2117, 2210, 2305

Offset: 2

Clark Kimberling

nonn

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 07 2012

Clark Kimberling, Table of n, a(n) for n = 2..200

Cf. A001000, A054060.

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[Pi/2 - ArcTan[h], {h, 1, 60}]]; leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

A024834 a(n) = least m such that if r and s in {1/1, 1/3, 1/5, ..., 1/(2n-1)} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

Original entry on oeis.org

4, 13, 26, 43, 64, 100, 133, 183, 226, 290, 343, 421, 484, 576, 676, 757, 871, 993, 1090, 1226, 1370, 1483, 1641, 1807, 1936, 2116, 2304, 2500, 2653, 2863, 3081, 3307, 3482, 3722, 3970, 4226, 4423, 4693, 4971, 5257, 5476, 5776, 6084, 6400, 6724, 6973, 7311, 7657, 8011

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024835.

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] - 1), #] &, Range[5]];
t[[2]] (* A024834 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A024835 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+1)/m < s for some integer k.

Original entry on oeis.org

7, 17, 31, 49, 81, 111, 157, 197, 257, 307, 381, 441, 529, 625, 703, 813, 931, 1025, 1157, 1297, 1407, 1561, 1723, 1849, 2025, 2209, 2401, 2551, 2757, 2971, 3193, 3365, 3601, 3845, 4097, 4291, 4557, 4831, 5113, 5329, 5625, 5929, 6241, 6561, 6807, 7141, 7483, 7833

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024834.

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]];
t[[2]] (* A024835 *)
(* Peter J. C. Moses, Aug 06 2012 *)

A024841 a(n) = least m such that if r and s in {1/1, 1/3, 1/5, ..., 1/(2n-1)} satisfy r < s, then r < k/m < (k+2)/m < s for some integer k.

Original entry on oeis.org

5, 19, 41, 71, 109, 155, 222, 287, 376, 460, 571, 673, 806, 926, 1081, 1219, 1396, 1552, 1751, 1926, 2147, 2380, 2584, 2839, 3106, 3338, 3627, 3928, 4188, 4511, 4846, 5134, 5491, 5860, 6176, 6567, 6970, 7385, 7740, 8177, 8626, 9087, 9481, 9964, 10459, 10966, 11398

Offset: 2

Clark Kimberling

nonn

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

Clark Kimberling, Table of n, a(n) for n = 2..100

Cf. A001000, A024842.

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]-1), #] &, Range[5]];
t[[3]] (* A024841 *)
(* Peter J. C. Moses, Aug 06 2012 *)

cp's OEIS Frontend

A214964 Least m > 0 such that for every r and s in the set S = {{h*(1+sqrt(5))/2} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k; m is the least separator of S as defined at A001000.

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A214965 Least m > 0 such that for every r and s in the set S = {{h*e} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k; m is the least separator of S as defined at A001000.

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A024838 Least m such that if r and s in {1/3, 1/6, 1/9, ..., 1/3n} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

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A024836 a(n) = least m such that if r and s in {1/1, 1/4, 1/7, ..., 1/(3n-2)} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

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A024837 a(n) = least m such that if r and s in {1/2, 1/5, 1/8, ..., 1/(3n-1)} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

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A024847 a(n) = least m such that if r and s in {1/1, 1/3, 1/5, ..., 1/(2n-1)} satisfy r < s, then r < k/m < (k+4)/m < s for some integer k.

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Mathematica

A024832 Least m such that if r and s in {Pi/2 - atn(h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k.

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A024834 a(n) = least m such that if r and s in {1/1, 1/3, 1/5, ..., 1/(2n-1)} satisfy r < s, then r < k/m < (k+1)/m < s for some integer k.

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Mathematica

A024835 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+1)/m < s for some integer k.

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