A001000 -id:A001000 - cp's OEIS frontend

A024822 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 < s for some integer k.

2, 5, 9, 22, 31, 53, 81, 97, 134, 177, 201, 253, 311, 342, 409, 482, 561, 603, 691, 785, 885, 937, 1046, 1161, 1282, 1409, 1475, 1611, 1753, 1901, 2055, 2215, 2297, 2466, 2641, 2822, 3009, 3202, 3301, 3503, 3711, 3925, 4145, 4371, 4486, 4721

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.

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[1/(3 h - 2), {h, 1, 60}]];
leastSeparator[t]

Corrected by Clark Kimberling, Aug 07 2012

A024823 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 < s for some integer k.

Original entry on oeis.org

3, 6, 17, 25, 45, 57, 86, 103, 141, 185, 209, 262, 321, 386, 421, 495, 575, 661, 706, 801, 902, 1009, 1122, 1181, 1303, 1431, 1565, 1705, 1777, 1926, 2081, 2242, 2409, 2495, 2671, 2853, 3041, 3235, 3435, 3537, 3746, 3961, 4182, 4409, 4642, 4881, 5003, 5251, 5505

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.

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[1/(3 h - 1), {h, 1, 60}]];
leastSeparator[t]

A024824 a(n) = least m such that if r and s in {1/3, 1/6, 1/9,..., 1/3n} satisfy r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

4, 7, 19, 28, 49, 61, 91, 127, 148, 193, 244, 271, 331, 397, 469, 508, 589, 676, 769, 817, 919, 1027, 1141, 1261, 1324, 1453, 1588, 1729, 1876, 1951, 2107, 2269, 2437, 2611, 2791, 2884, 3073, 3268, 3469, 3676, 3889, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5548

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..300

Cf. A001000.

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[1/(3 n), {n, 1, 60}]];
leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

A024825 a(n) = least m such that if r and s in {1/4, 1/8, 1/12,..., 1/4n} satisfy r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

5, 9, 25, 37, 65, 81, 121, 169, 197, 257, 325, 361, 441, 529, 625, 677, 785, 901, 1025, 1089, 1225, 1369, 1521, 1681, 1765, 1937, 2117, 2305, 2501, 2601, 2809, 3025, 3249, 3481, 3721, 3845, 4097, 4357, 4625, 4901, 5185, 5329, 5625, 5929, 6241

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..300

Cf. A001000.

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[1/(4 h), {h, 1, 60}]];
leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

Corrected and edited by Clark Kimberling, Aug 07 2012

A024826 Least m such that if r and s in {1/1, 1/3, 1/6,..., 1/C(n+1,2)} satisfy r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

2, 4, 7, 13, 31, 46, 64, 85, 145, 181, 226, 331, 397, 469, 638, 736, 841, 1089, 1225, 1378, 1711, 1901, 2311, 2542, 2784, 3313, 3601, 3901, 4564, 4915, 5685, 6091, 6526, 7441, 7937, 8977, 9538, 10116, 11341, 11989, 13358, 14080, 14821, 16401, 17221, 18964, 19867

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..100

Cf. A001000.

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[1/Binomial[h + 1, 2], {h, 1, 50}]]
leastSeparator[t]

A024827 Least m such that if r and s in {1/1, 1/4, 1/9,..., 1/n^2} satisfy r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

2, 5, 10, 19, 33, 76, 109, 148, 197, 325, 406, 501, 727, 865, 1015, 1373, 1576, 1801, 2313, 2602, 3250, 3611, 4001, 4852, 5325, 5820, 6913, 7501, 8789, 9478, 10207, 11775, 12616, 14416, 15377, 16385, 18514, 19653, 22051, 23329, 24643, 27437, 28900, 32001, 33621

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..100

Cf. A001000.

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[1/h^2, {h, 1, 60}]]
leastSeparator[t]

A024828 a(n) = least m such that if r and s in {h/(1 + h^2): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k.

Original entry on oeis.org

7, 9, 11, 14, 18, 27, 32, 44, 58, 66, 83, 102, 112, 134, 158, 184, 198, 227, 258, 291, 308, 344, 382, 422, 464, 486, 531, 578, 627, 678, 704, 758, 814, 872, 932, 994, 1026, 1091, 1158, 1227, 1298, 1371, 1408, 1484, 1562, 1642, 1724, 1808, 1894, 1938, 2027, 2118, 2211, 2306

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.

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[h/(1 + h^2), {h, 1, 60}]]
leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

A024829 a(n) = least m such that if r and s in {F(2h-1)/F(2h): 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

4, 11, 29, 173, 1063, 7074, 47753, 325961, 2228269, 15262701, 104577551, 716721983, 4912208209

Offset: 2

Clark Kimberling

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

Cf. A001000, A024830.

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[2 h - 1]/Fibonacci[2 h]], {h, 1, 10}]
t1 = leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

Corrected by Clark Kimberling, Aug 07 2012

a(11)-a(14) from Sean A. Irvine, Jul 25 2019

A024830 a(n) = least m such that if r and s in {F(2h)/F(2h+1): 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

7, 18, 73, 424, 2741, 18389, 124799, 851937, 5831634, 39952039, 273777171, 1876334786, 12860231668

Offset: 2

Clark Kimberling

See A001000 for a guide to related sequences. - Clark Kimberling, Aug 07 2012

Cf. A001000, A024829.

(* For a guide to related sequences, see A001000. *)
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[2 h]/Fibonacci[2 h + 1]], {h, 1, 10}];
t1 = leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

Extended by Clark Kimberling, Aug 07 2012

a(11)-a(14) from Sean A. Irvine, Jul 25 2019

A024833 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+1)/m < s for some integer k.

Original entry on oeis.org

5, 11, 19, 29, 41, 61, 79, 106, 129, 163, 191, 232, 265, 313, 365, 407, 466, 529, 579, 649, 723, 781, 862, 947, 1013, 1105, 1201, 1301, 1379, 1486, 1597, 1712, 1801, 1923, 2049, 2179, 2279, 2416, 2557, 2702, 2813, 2965, 3121, 3281, 3445, 3571, 3742, 3917, 4096

Offset: 2

Clark Kimberling

nonn

For a guide to related sequences, see A001000. - Peter J. C. Moses, Aug 08 2012

			Using the terminology introduced at A001000, the 2nd separator of the set {1/3, 1/2, 1} is a(3) = 11, since 1/3 < 4/11 < 5/11 < 1/2 < 6/11 < 7/11 < 1 and 11 is the least m for which 1/3, 1/2, 1 are thus separated using numbers k/m. - _Clark Kimberling_, Aug 08 2012

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

Cf. A001000, A071111, A024843, A024846.

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[[2]] (* Clark Kimberling, Aug 08 2012 *)

cp's OEIS Frontend

A024822 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 < s for some integer k.

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A024823 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 < s for some integer k.

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A024824 a(n) = least m such that if r and s in {1/3, 1/6, 1/9,..., 1/3n} satisfy r < s, then r < k/m < s for some integer k.

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A024825 a(n) = least m such that if r and s in {1/4, 1/8, 1/12,..., 1/4n} satisfy r < s, then r < k/m < s for some integer k.

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A024826 Least m such that if r and s in {1/1, 1/3, 1/6,..., 1/C(n+1,2)} satisfy r < s, then r < k/m < s for some integer k.

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A024827 Least m such that if r and s in {1/1, 1/4, 1/9,..., 1/n^2} satisfy r < s, then r < k/m < s for some integer k.

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A024828 a(n) = least m such that if r and s in {h/(1 + h^2): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k.

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A024829 a(n) = least m such that if r and s in {F(2*h-1)/F(2*h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

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Extensions

A024830 a(n) = least m such that if r and s in {F(2*h)/F(2*h+1): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

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A024829 a(n) = least m such that if r and s in {F(2h-1)/F(2h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

A024830 a(n) = least m such that if r and s in {F(2h)/F(2h+1): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).