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 1-7 of 7 results.

A361848 Number of integer partitions of n such that (maximum) <= 2*(median).

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 12, 15, 19, 26, 31, 40, 49, 61, 75, 93, 112, 137, 165, 199, 238, 289, 341, 408, 482, 571, 674, 796, 932, 1096, 1280, 1495, 1738, 2026, 2347, 2724, 3148, 3639, 4191, 4831, 5545, 6372, 7298, 8358, 9552, 10915, 12439, 14176, 16121, 18325
Offset: 0

Views

Author

Gus Wiseman, Mar 28 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (321)     (331)
                                     (2211)    (421)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (22111)
                                               (211111)
                                               (1111111)
For example, the partition y = (3,2,2) has maximum 3 and median 2, and 3 <= 2*2, so y is counted under a(7).

Crossrefs

For length instead of median we have A237755.
For minimum instead of median we have A237824.
The equal case is A361849, ranks A361856.
For mean instead of median we have A361851.
The complement is counted by A361857, ranks A361867.
The unequal case is A361858.
Reversing the inequality gives A361859, ranks A361868.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A061395, A067538, A111907, A240219, A324562, A359907, A361394, A361860.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#<=2*Median[#]&]],{n,30}]

Formula

a(n) = A361849(n) + A361858(n).
a(n) = A000041(n) - A361857(n).

A361858 Number of integer partitions of n such that the maximum is less than twice the median.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 8, 12, 15, 19, 22, 31, 34, 45, 55, 67, 78, 100, 115, 144, 170, 203, 238, 291, 337, 403, 473, 560, 650, 772, 889, 1046, 1213, 1414, 1635, 1906, 2186, 2533, 2913, 3361, 3847, 4433, 5060, 5808, 6628, 7572, 8615, 9835, 11158, 12698, 14394
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (322)      (71)
                                     (321)     (331)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
The partition y = (3,2,2,1) has maximum 3 and median 2, and 3 < 2*2, so y is counted under a(8).

Crossrefs

For minimum instead of median we have A053263.
For length instead of median we have A237754.
Allowing equality gives A361848, strict A361850.
The equal version is A361849, ranks A361856.
For mean instead of median we have A361852.
Reversing the inequality gives A361857, ranks A361867.
The complement is counted by A361859, ranks A361868.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A237751, A237755, A237820, A237824, A240219, A361394, A361851, A361860, A361907.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#<2*Median[#]&]],{n,30}]

A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 10, 15, 23, 34, 46, 67, 90, 121, 164, 219, 285, 375, 483, 622, 799, 1017, 1284, 1621, 2033, 2537, 3158, 3915, 4832, 5953, 7303, 8930, 10896, 13248, 16071, 19451, 23482, 28272, 33977, 40736, 48741, 58201, 69367, 82506, 97986, 116139
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(4) = 1 through a(9) = 15 partitions:
  (211)  (311)   (411)    (421)     (422)      (522)
         (2111)  (3111)   (511)     (521)      (621)
                 (21111)  (3211)    (611)      (711)
                          (4111)    (4211)     (4221)
                          (22111)   (5111)     (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (33111)
                                    (311111)   (42111)
                                    (2111111)  (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).

Crossrefs

For length instead of median we have A237752.
For minimum instead of median we have A237821.
Reversing the inequality gives A361848.
The equal case is A361849, ranks A361856.
The unequal case is A361857, ranks A361867.
The complement is counted by A361858.
These partitions have ranks A361868.
For mean instead of median we have A361906.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#>=2*Median[#]&]],{n,30}]

A361857 Number of integer partitions of n such that the maximum is greater than twice the median.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 7, 11, 16, 25, 37, 52, 74, 101, 138, 185, 248, 325, 428, 554, 713, 914, 1167, 1476, 1865, 2336, 2922, 3633, 4508, 5562, 6854, 8405, 10284, 12536, 15253, 18489, 22376, 26994, 32507, 39038, 46802, 55963, 66817, 79582, 94643, 112315
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(5) = 1 through a(10) = 16 partitions:
  (311)  (411)   (511)    (521)     (522)      (622)
         (3111)  (4111)   (611)     (621)      (721)
                 (31111)  (4211)    (711)      (811)
                          (5111)    (5211)     (5221)
                          (32111)   (6111)     (5311)
                          (41111)   (33111)    (6211)
                          (311111)  (42111)    (7111)
                                    (51111)    (43111)
                                    (321111)   (52111)
                                    (411111)   (61111)
                                    (3111111)  (331111)
                                               (421111)
                                               (511111)
                                               (3211111)
                                               (4111111)
                                               (31111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 > 2*2, so y is counted under a(10).

Crossrefs

For length instead of median we have A237751.
For minimum instead of median we have A237820.
The complement is counted by A361848.
The equal version is A361849, ranks A361856.
Reversing the inequality gives A361858.
Allowing equality gives A361859, ranks A361868.
These partitions have ranks A361867.
For mean instead of median we have A361907.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A061395, A237755, A237824, A240219, A361394, A361851, A361860.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#>2*Median[#]&]],{n,30}]

A361868 Positive integers > 1 whose prime indices satisfy (maximum) >= 2*(median).

Original entry on oeis.org

12, 20, 24, 28, 40, 42, 44, 48, 52, 56, 60, 63, 66, 68, 72, 76, 78, 80, 84, 88, 92, 96, 99, 102, 104, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 144, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 189, 190, 192, 195
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 >= 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}

Crossrefs

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal case is A361856, counted by A361849.
These partitions are counted by A361859.
The unequal case is A361867, counted by A361857.
The complement is counted by A361858.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A053263, A111907, A118096, A237753, A237821, A361848, A361855, A361906, A361908.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]>=2*Median[prix[#]]&]

A361850 Number of strict integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 3, 3, 4, 2, 5, 4, 7, 8, 10, 6, 11, 11, 15, 16, 21, 18, 25, 23, 28, 32, 40, 40, 51, 51, 58, 60, 73, 75, 93, 97, 113, 123, 139, 141, 164, 175, 199, 217, 248, 263, 301, 320, 356, 383, 426, 450, 511, 551, 613, 664, 737
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(7) = 1 through a(20) = 4 strict partitions (A..C = 10..12):
  421  .  .  631  632   .  841   842  843   A51    A52    A53   A54   C62
                  5321     6421       7431  7432   8531   8532  C61   9542
                                      7521  64321  8621         9541  9632
                                                   65321        9631  85421
                                                                9721
The partition (7,4,3,1) has maximum 7 and median 7/2, so is counted under a(15).
The partition (8,6,2,1) has maximum 8 and median 4, so is counted under a(17).

Crossrefs

For minimum instead of median we have A241035, non-strict A237824.
For length instead of median we have A241087, non-strict A237755.
The non-strict version is A361849, ranks A361856.
The non-strict complement is counted by A361857, ranks A361867.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A359907 counts strict partitions with integer median
A360005 gives median of prime indices (times two), distinct A360457.
Cf. A027193, A067659, A079309, A111907, A116608, A359897, A359908, A360952, A361851, A361858, A361859, A361860.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#==2*Median[#]&]],{n,30}]

A362048 Number of integer partitions of n such that (length) <= 2*(median).

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 8, 12, 15, 20, 25, 33, 41, 53, 66, 85, 105, 134, 164, 205, 250, 308, 373, 456, 549, 666, 799, 963, 1152, 1382, 1645, 1965, 2330, 2767, 3269, 3865, 4546, 5353, 6274, 7357, 8596, 10046, 11700, 13632, 15834, 18394, 21312, 24690, 28534, 32974
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 1 through a(9) = 15 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)  (32)   (33)   (43)    (44)    (54)
                   (31)  (41)   (42)   (52)    (53)    (63)
                         (221)  (51)   (61)    (62)    (72)
                                (222)  (322)   (71)    (81)
                                (321)  (331)   (332)   (333)
                                       (421)   (422)   (432)
                                       (2221)  (431)   (441)
                                               (521)   (522)
                                               (2222)  (531)
                                               (3221)  (621)
                                               (3311)  (3222)
                                                       (3321)
                                                       (4221)
                                                       (4311)

Crossrefs

For maximum instead of median we have A237755.
For minimum instead of median we have A237800.
For maximum instead of length we have A361848.
The equal case is A362049.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A237824, A240219, A361394, A361851, A361856-A361860, A361867, A361868.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]<=2*Median[#]&]],{n,30}]
Showing 1-7 of 7 results.

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