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.

Showing 1-8 of 8 results.

A241089 Number of partitions p of n into distinct parts such that max(p) > 2*(number of parts of p).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, 24, 29, 35, 42, 50, 61, 72, 85, 101, 118, 138, 161, 188, 218, 254, 293, 339, 391, 450, 515, 591, 675, 771, 878, 999, 1135, 1289, 1460, 1652, 1868, 2108, 2376, 2676, 3009, 3379, 3793, 4250, 4760, 5325, 5952
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(9) counts these 5 partitions:  9, 81, 72, 63, 54.

Crossrefs

Cf. A241085, A241086, A241087, A241088.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241087 Number of partitions p of n into distinct parts such that max(p) = 2*(number of parts of p).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 6, 5, 6, 6, 7, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20, 23, 25, 28, 30, 35, 38, 43, 46, 51, 55, 61, 64, 72, 76, 84, 91, 101, 109, 120, 130, 142, 155, 168, 181, 196, 212, 228, 248, 266, 288, 311, 337
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(15) counts these 2 partitions:  8421, 654.

Crossrefs

Cf. A241085, A241086, A241088, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241085 Number of partitions p of n into distinct parts such that max(p) < 2*(number of parts of p).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 10, 11, 13, 14, 17, 18, 21, 22, 25, 27, 31, 33, 38, 42, 47, 52, 57, 63, 69, 76, 82, 91, 99, 109, 119, 132, 142, 158, 171, 188, 203, 223, 240, 263, 284, 309, 334, 364, 393, 428, 463, 501, 543, 588
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(15) counts these 5 partitions:  7521, 7431, 6531, 6432, 54321.

Crossrefs

Cf. A241086, A241087, A241088, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241088 Number of partitions p of n into distinct parts such that max(p) >= 2*(number of parts of p).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 4, 6, 7, 10, 12, 15, 18, 22, 26, 32, 39, 46, 56, 66, 78, 91, 108, 125, 147, 171, 200, 231, 269, 309, 357, 410, 470, 538, 616, 703, 801, 913, 1037, 1178, 1335, 1511, 1707, 1929, 2172, 2448, 2752, 3093, 3470, 3894, 4359, 4880, 5455
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(9) counts these 6 partitions:  9, 81, 72, 63, 621, 54.

Crossrefs

Cf. A241085, A241086, A241087, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241091 Number of partitions p of n into distinct parts such that max(p) <= 1 + 2*(number of parts of p).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 4, 5, 5, 7, 7, 9, 10, 11, 12, 15, 16, 19, 22, 24, 27, 30, 34, 37, 43, 47, 53, 59, 66, 72, 82, 88, 99, 109, 120, 131, 146, 160, 176, 194, 212, 233, 256, 279, 304, 334, 362, 396, 431, 471, 510, 558, 604, 659, 714, 776, 839, 913, 985
Offset: 0

Views

Author

Clark Kimberling, Apr 18 2014

Keywords

Examples

			a(12) counts these 5 partitions:  741, 732, 651, 642, 6321, 543, 5421.

Crossrefs

Cf. A241086, A241092, A241093.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
 Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
 Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
 Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
 Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
 Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)

Formula

a(n) = A241086(n) + A241092(n) for n >= 0.

A241092 Number of partitions p of n into distinct parts such that max(p) = 1 + 2*(number of parts of p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7, 7, 8, 9, 10, 10, 12, 13, 15, 17, 19, 21, 25, 26, 29, 32, 35, 38, 42, 46, 51, 57, 62, 69, 76, 83, 90, 100, 107, 117, 127, 139, 150, 165, 178, 195, 212, 231, 250, 273, 294, 319, 346, 373, 402
Offset: 0

Views

Author

Clark Kimberling, Apr 18 2014

Keywords

Examples

			a(12) counts these 5 partitions:  741, 732, 651, 642, 6321, 543, 5421.

Crossrefs

Cf. A241086, A241091, A241093, A000009.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)

Formula

a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.
a(n) = A241091(n) - A241086(n) for n >= 0.

A241093 Number of partitions p of n into distinct parts such that max(p) > 1 + 2*(number of parts of p).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 21, 26, 31, 38, 45, 54, 65, 77, 92, 108, 128, 149, 175, 203, 237, 274, 318, 366, 424, 486, 559, 640, 733, 836, 953, 1084, 1232, 1398, 1583, 1792, 2025, 2286, 2576, 2902, 3262, 3666, 4111, 4610, 5160, 5774
Offset: 0

Views

Author

Clark Kimberling, Apr 18 2014

Keywords

Examples

			a(12) counts these 8 partitions: {12}, {11,1}, {10,2}, {9,3}, {9,2,1}, {8,4}, {8,3,1}, {7,5}.

Crossrefs

Cf. A241086, A241091, A241092, A000009.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)

Formula

a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.

A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014
Offset: 1

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Also strict partitions such that (maximum) <= 2*(mean).
These are strict partitions whose complement (see A361851) has size <= n.

Examples

			The partition y = (4,3,1) has length 3 and maximum 4, and 3*4 <= 2*8, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.

Crossrefs

The equal case for median is A361850, non-strict A361849 (ranks A361856).
The non-strict version is A361851, A361848 for median.
The equal case is A361854, non-strict A361853 (ranks A361855).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]
Showing 1-8 of 8 results.

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