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.

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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}]

A363133 Numbers > 1 whose prime indices satisfy 2*(minimum) = (mean).

Original entry on oeis.org

10, 28, 30, 39, 84, 88, 90, 100, 115, 171, 208, 252, 255, 259, 264, 270, 273, 280, 300, 363, 517, 544, 624, 756, 783, 784, 792, 793, 810, 840, 880, 900, 925, 1000, 1035, 1085, 1197, 1216, 1241, 1425, 1495, 1521, 1595, 1615, 1632, 1683, 1691, 1785, 1872, 1911
Offset: 1

Views

Author

Gus Wiseman, May 29 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.

Examples

			The terms together with their prime indices begin:
    10: {1,3}
    28: {1,1,4}
    30: {1,2,3}
    39: {2,6}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}
   115: {3,9}
   171: {2,2,8}
   208: {1,1,1,1,6}
   252: {1,1,2,2,4}
   255: {2,3,7}
   259: {4,12}
   264: {1,1,1,2,5}

Crossrefs

Removing the factor 2 gives A000961.
For maximum instead of mean we have A361908, counted by A118096.
Partitions of this type are counted by A363132.
For length instead of mean we have A363134, counted by A237757.
For 2*(maximum) = (length) we have A363218, counted by A237753.
A051293 counts subsets with integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Cf. A006141, A106529, A111907, A237755, A237824, A324522, A327482, A361860, A361861, A362050.

Programs

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

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2023

Keywords

Comments

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

Examples

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.

Crossrefs

Positions of zeros are 1 and A000040.
For length instead of mean we have A237832.
For minimum instead of mean we have A118096.
These partitions have ranks A362047.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean.
A097364 counts partitions by (maximum) - (minimum).
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]
Previous Showing 21-23 of 23 results.

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