A361848 -id:A361848 - cp's OEIS frontend

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

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

Gus Wiseman, Apr 02 2023

nonn

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

			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).

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.

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

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234

Offset: 1

Gus Wiseman, Apr 05 2023

nonn

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).

			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:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

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

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

Gus Wiseman, Apr 05 2023

nonn

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).

			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}

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.

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

A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 5, 0, 6, 3, 5, 0, 11, 6, 8, 7, 10, 0, 36, 0, 14, 16, 16, 29, 43, 0, 21, 36, 69, 0, 97, 0, 35, 138, 33, 0, 150, 61, 137, 134, 74, 0, 231, 134, 265, 229, 56, 0, 650, 0, 65, 749, 267, 247, 533, 0, 405, 565

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also strict partitions satisfying (maximum) = 2*(mean).

These are strict partitions where both the diagram and its complement (see example) have size n.

			The a(n) strict partitions for selected n (A..E = 10..14):
  n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
--------------------------------------------------------------
  621   831    7421   A32    8431   C42    A532   E43    B542
        6321          A41    8521   C51    A541   E52    B632
                                    9432   A631   E61    B641
                                    9531   A721          B731
                                    9621   85421         B821
                                           86321
The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
The strict partition y = (8,5,4,2,1) has diagram:
  o o o o o o o o
  o o o o o . . .
  o o o o . . . .
  o o . . . . . .
  o . . . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(20).

For minimum instead of mean we have A241035, non-strict A118096.
For length instead of mean we have A241087, non-strict A237753.
For median instead of mean we have A361850, non-strict A361849.
The non-strict version is A361853.
These partitions have ranks A361855 /\ A005117.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A008289 counts strict partitions by length.
A102627 counts strict partitions with integer mean, non-strict A067538.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

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

A361861 Number of integer partitions of n where the median is twice the minimum.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 5, 8, 11, 16, 20, 28, 38, 53, 67, 87, 111, 146, 183, 236, 297, 379, 471, 591, 729, 909, 1116, 1376, 1682, 2065, 2507, 3055, 3699, 4482, 5395, 6501, 7790, 9345, 11153, 13316, 15839, 18844, 22333, 26466, 31266, 36924, 43478, 51177

Offset: 1

Gus Wiseman, Apr 02 2023

nonn

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

			The a(4) = 1 through a(11) = 11 partitions:
  (31)  (221)  (321)  (421)   (62)     (621)    (442)     (542)
                      (2221)  (521)    (4221)   (721)     (821)
                              (3221)   (4311)   (5221)    (6221)
                              (3311)   (22221)  (5311)    (6311)
                              (22211)  (32211)  (32221)   (33221)
                                                (33211)   (42221)
                                                (42211)   (43211)
                                                (222211)  (52211)
                                                          (222221)
                                                          (322211)
                                                          (2222111)
The partition (3,2,2,2,1,1) has median 2 and minimum 1, so is counted under a(11).
The partition (5,4,2) has median 4 and minimum 2, so is counted under a(11).

For maximum instead of median we have A118096.
For length instead of median we have A237757, without the coefficient A006141.
With minimum instead of twice minimum we have A361860.
A000041 counts integer partitions, strict A000009.
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.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A039900, A053263, A067659, A111907, A116608, A237753, A237755, A237824, A361848, A361853.

Table[Length[Select[IntegerPartitions[n],2*Min@@#==Median[#]&]],{n,30}]

A362049 Number of integer partitions of n such that (length) = 2*(median).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 3, 3, 3, 3, 3, 3, 4, 5, 9, 12, 19, 22, 29, 32, 39, 43, 51, 57, 70, 81, 101, 123, 153, 185, 230, 272, 328, 386, 454, 526, 617, 708, 824, 951, 1106, 1277, 1493, 1727, 2020, 2344, 2733, 3164, 3684, 4245, 4914, 5647, 6502, 7438, 8533, 9730

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). All of these partitions have even length, because an odd-length multiset cannot have fractional median.

			The a(13) = 3 through a(15) = 5 partitions:
  (7,2,2,2)  (8,2,2,2)      (9,2,2,2)
  (8,2,2,1)  (9,2,2,1)      (10,2,2,1)
  (8,3,1,1)  (9,3,1,1)      (10,3,1,1)
             (3,3,3,3,1,1)  (3,3,3,3,2,1)
                            (4,3,3,3,1,1)

For maximum instead of median we have A237753.
For minimum instead of median we have A237757.
For maximum instead of length we have A361849, ranks A361856.
This is the equal case of A362048.
These partitions have ranks A362050.
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, A079309, A237800, A240219, A361801, A361848, A361858, A361859.

Table[Length[Select[IntegerPartitions[n],Length[#]==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

Gus Wiseman, Apr 10 2023

nonn

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

			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)

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.

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

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

Gus Wiseman, May 23 2023

nonn

Also strict partitions such that (maximum) <= 2*(mean).

These are strict partitions whose complement (see A361851) has size <= n.

			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.

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.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]

cp's OEIS Frontend

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

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A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

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

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A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

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A361861 Number of integer partitions of n where the median is twice the minimum.

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A362049 Number of integer partitions of n such that (length) = 2*(median).

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

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

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