A359897 -id:A359897 - cp's OEIS frontend

A307683 Number of partitions of n having a non-integer median.

0, 0, 1, 0, 2, 1, 4, 1, 7, 5, 11, 8, 18, 17, 31, 28, 47, 51, 75, 81, 119, 134, 181, 206, 277, 323, 420, 488, 623, 737, 922, 1084, 1352, 1597, 1960, 2313, 2819, 3330, 4029, 4743, 5704, 6722, 8030, 9434, 11234, 13175, 15601, 18262, 21552, 25184, 29612, 34518

Offset: 1

Clark Kimberling, Apr 24 2019

This sequence and A325347 partition the partition numbers, A000041.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). - Gus Wiseman, Mar 16 2023

			a(7) counts these 4 partitions: [6,1], [5,2], [4,3], [3,2,1,1].

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..180

The complement is counted by A325347, strict A359907.
For mean instead of median we have A349156, strict A361391.
These partitions have ranks A359912, complement A359908.
The strict case is A360952.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A359893/A359901/A359902 count partitions by median.
Cf. A000016, A051293, A067538, A082550, A240219, A240850, A316413, A326567/A326568, A327475, A359897, A360005.

Table[Count[IntegerPartitions[n], q_ /; !IntegerQ[Median[q]]], {n, 10}]

A359907 Number of strict integer partitions of n with integer median.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 4, 2, 6, 4, 9, 6, 14, 10, 18, 16, 27, 23, 36, 34, 51, 49, 67, 68, 94, 95, 122, 129, 166, 174, 217, 233, 287, 308, 371, 405, 487, 528, 622, 683, 805, 880, 1024, 1127, 1305, 1435, 1648, 1818, 2086, 2295, 2611, 2882, 3273, 3606, 4076, 4496, 5069

Offset: 0

Gus Wiseman, Jan 21 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(14) = 18 partitions (A..E = 10..14):
  1  2  3  4   5  6    7    8    9    A    B    C     D     E
           31     42   421  53   432  64   542  75    643   86
                  51        62   531  73   632  84    652   95
                  321       71   621  82   641  93    742   A4
                            431       91   731  A2    751   B3
                            521       532  821  B1    832   C2
                                      541       543   841   D1
                                      631       642   931   653
                                      721       651   A21   743
                                                732   6421  752
                                                741         761
                                                831         842
                                                921         851
                                                5421        932
                                                            941
                                                            A31
                                                            B21
                                                            7421

For mean instead of median: A102627, non-strict A067538 (ranked by A316413).
This is the strict case of A325347, ranked by A359908.
The median statistic is ranked by A360005(n)/2.
A000041 counts partitions, strict A000009.
A051293 counts subsets with integer mean, median A000975, cf. A005578.
A058398 counts partitions by mean, see also A008284, A327482.
A326567/A326568 gives the mean of prime indices.
A359893, A359901, A359902 count partitions by median.
Cf. A000016, A082550, A240219, A240850, A326669, A327475, A328966, A359897.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[Median[#]]&]],{n,0,30}]

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Gus Wiseman, Jan 22 2023

nonn

First differs from A236510 in having 252 (prime indices {1,1,2,2,4}).
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 900 are {1,1,2,2,3,3}, with mean 2 and median 2, so 900 is in the sequence.

These partitions are counted by A240219, strict A359897.
The LHS (mean of prime indices) is A326567/A326568.
The complement is A359890, counted by A359894.
The odd-length case is A359891, complement A359892, counted by A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359893 and A359901 count partitions by median, odd-length A359902.
A359908 lists numbers whose prime indices have integer median.
Cf. A026424, A327473, A359899, A359903, A359905, A359909, A360006, A360007, A360008, A360009.

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

Numbers n such that A326567(n)/A326568(n) = A360005(n)/2.

A359894 Number of integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 10, 13, 20, 28, 49, 53, 93, 113, 145, 203, 287, 329, 479, 556, 724, 955, 1242, 1432, 1889, 2370, 2863, 3502, 4549, 5237, 6825, 8108, 9839, 12188, 14374, 16958, 21617, 25852, 30582, 36100, 44561, 51462, 63238, 73386, 85990, 105272, 124729

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(4) = 1 through a(8) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)
         (311)   (3111)   (331)     (422)
         (2111)  (21111)  (421)     (431)
                          (511)     (521)
                          (2221)    (611)
                          (3211)    (4211)
                          (4111)    (5111)
                          (22111)   (22211)
                          (31111)   (32111)
                          (211111)  (41111)
                                    (221111)
                                    (311111)
                                    (2111111)

The complement is counted by A240219.
These partitions are ranked by A359890, complement A359889.
The odd-length case is ranked by A359892, complement A359891.
The odd-length case is A359896, complement A359895.
The strict case is A359898, complement A359897.
The odd-length strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284 and A058398 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
A359909 counts factorizations with the same mean as median, odd-len A359910.
Cf. A066571, A316313, A327472, A327482, A349156, A359906.

Table[Length[Select[IntegerPartitions[n],Mean[#]!=Median[#]&]],{n,0,30}]

A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 6, 0, 0, 0, 6, 0, 7, 0, 1, 0, 0, 0, 0, 90, 0, 63, 0, 0, 0, 0, 11, 0, 0, 0, 436, 0, 0, 0, 0, 0, 0, 0, 0, 2157, 0, 0, 240, 1595, 22, 0, 0, 0, 6464, 0, 0, 0, 0, 0, 0, 0, 0, 11628, 4361, 0, 0, 0, 0, 0, 0, 0, 12927, 0, 0, 621, 0

Offset: 0

Gus Wiseman, Jan 27 2023

nonn

Note that such a partition cannot be strict for n > 1.

Conjecture: If n is squarefree, then a(n) = 0.

			The n = 1, 4, 8, 9, 12, 16, 18 partitions (D=13):
  (1)  (22)  (3311)  (333)  (322221)  (4444)      (444222)
             (5111)         (332211)  (43222111)  (444411)
                            (422211)  (43321111)  (552222)
                            (522111)  (53221111)  (555111)
                            (531111)  (54211111)  (771111)
                            (621111)  (63211111)  (822222)
                                                  (D11111)
For example, the partition (4,3,3,3,3,3,2,2,1,1) has mean 5/2, and its multiplicities (1,5,2,2) also have mean 5/2, so it is counted under a(20).

These partitions are ranked by A359903, for prime factors A359904.
Positions of positive terms are A360070.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
A360069 counts partitions whose multiplicities have integer mean.
Cf. A067340, A067538, A082550, A240219, A316313, A327475, A349156, A359893, A359897, A359905.

Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Length/@Split[#]]&]],{n,0,30}]

A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 46, 61, 82, 102, 136, 174, 224, 283, 360, 449, 569, 708, 883, 1089, 1352, 1659, 2042, 2492, 3039, 3695, 4492, 5426, 6555, 7889, 9482, 11360, 13602, 16231, 19348, 23005, 27313, 32364, 38303, 45227, 53341, 62800, 73829

Offset: 1

Gus Wiseman, Jun 26 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).

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (3221)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For mean instead of mode we have A240219, see A359894, A359889, A359895, A359897, A359899.
Including mean also gives A363719, ranks A363727.
For mean instead of median we have A363723, see A363724, A363731.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode, ranks A356862.
Cf. A027193, A237984, A325347, A362562, A363720, A363725, A363726.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Median[#]}==modes[#]&]],{n,30}]

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020

Offset: 1

Gus Wiseman, Jun 19 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).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
Without loss of generality, we may assume there is a unique middle-part (A238478).
Includes all constant partitions.

			The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
  1  2   4     6       8         C             E               G
     11  22    33      44        66            77              88
         1111  222     2222      444           2222222         4444
               111111  3221      3333          3222221         5443
                       11111111  4332          3322211         6442
                                 5331          4222211         7441
                                 222222        11111111111111  22222222
                                 322221                        32222221
                                 422211                        33222211
                                 111111111111                  42222211
                                                               52222111
                                                               1^16

For unequal instead of equal: A363720, ranks A363730, unique mode A363725.
The odd-length case is A363721.
These partitions have ranks A363727, nonprime A363722.
The case of non-constant partitions is A363728, ranks A363729.
The version for factorizations is A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363742.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 1, 5, 5, 2, 5, 2, 8, 18, 1, 2, 19, 2, 24, 41, 20, 2, 9, 44, 31, 94, 102, 2, 125, 2, 1, 206, 68, 365, 382, 2, 98, 433, 155, 2, 716, 2, 1162, 2332, 196, 2, 17, 1108, 563, 1665, 3287, 2, 3906, 5474, 2005, 3083, 509, 2, 9029

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

The length and median of such a partition are integers with product n.

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)  (9)
            (111)       (11111)  (222)  (1111111)       (333)
                                 (321)                  (432)
                                                        (531)
                                                        (111111111)
The a(15) = 18 partitions:
  (15)
  (5,5,5)
  (6,5,4)
  (7,5,3)
  (8,5,2)
  (9,5,1)
  (3,3,3,3,3)
  (4,3,3,3,2)
  (4,4,3,2,2)
  (4,4,3,3,1)
  (5,3,3,2,2)
  (5,3,3,3,1)
  (5,4,3,2,1)
  (5,5,3,1,1)
  (6,3,3,2,1)
  (6,4,3,1,1)
  (7,3,3,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

This is the odd-length case of A240219, complement A359894, strict A359897.
These partitions are ranked by A359891, complement A359892.
The complement is counted by A359896.
The strict case is A359899, complement A359900.
The version for factorizations is A359910.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008289, A316313, A327472, A327475, A327482, A359889, A359906.

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]

\\ P(n, k, m) is g.f. for k parts of max size m.
P(n, k, m)={polcoef(1/prod(i=1, m, 1 - y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ Andrew Howroyd, Jan 21 2023

a(p) = 2 for prime p. - Andrew Howroyd, Jan 21 2023

A363720 Number of integer partitions of n with different mean, median, and mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 5, 7, 16, 17, 34, 38, 50, 79, 115, 123, 198, 220, 291, 399, 536, 605, 815, 1036, 1241, 1520, 2059, 2315, 3132, 3708, 4491, 5668, 6587, 7788, 10259, 12299, 14515, 17153, 21558, 24623, 30876, 35540, 41476, 52023, 61931, 70811, 85545

Offset: 0

Gus Wiseman, Jun 21 2023

nonn

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(7) = 2 through a(11) = 16 partitions:
  (421)   (431)   (621)    (532)     (542)
  (3211)  (521)   (3321)   (541)     (632)
          (4211)  (4311)   (631)     (641)
                  (5211)   (721)     (731)
                  (32211)  (5311)    (821)
                           (6211)    (4322)
                           (322111)  (4421)
                                     (5321)
                                     (5411)
                                     (6311)
                                     (7211)
                                     (33221)
                                     (43211)
                                     (52211)
                                     (332111)
                                     (422111)

For equal instead of unequal: A363719, ranks A363727, odd-length A363721.
The case of a unique mode is A363725.
These partitions have ranks A363730.
For factorizations we have A363742, for equal A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363728.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,0,30}]

A359910 Number of odd-length integer factorizations of n into factors > 1 with the same mean as median.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Gus Wiseman, Jan 24 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(n) factorizations for n = 120, 960, 5760, 6720:
  120      960         5760            6720
  4*5*6    2*16*30     16*18*20        4*30*56
  2*6*10   4*12*20     3*5*6*8*8       10*21*32
           8*10*12     4*4*6*6*10      12*20*28
           3*4*4*4*5   2*2*8*10*18     4*5*6*7*8
                       2*2*2*4*4*5*9   2*4*7*10*12
                                       2*2*2*4*5*6*7

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A359895, ranked by A359891.
This is the odd-length case of A359909, partitions A240219.
A001055 counts factorizations.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A316313, A326567/A326568, A359889, A359894, A359897, A359902, A359906, A359911, A360005.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,100}]

A359910(n, m=n, facs=List([])) = if(1==n, (((#facs)%2) && (facs[(1+#facs)/2]==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359910(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

More terms from Antti Karttunen, Jan 20 2025

cp's OEIS Frontend

A307683 Number of partitions of n having a non-integer median.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A359907 Number of strict integer partitions of n with integer median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A359894 Number of integer partitions of n whose parts do not have the same mean as median.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A363720 Number of integer partitions of n with different mean, median, and mode.

Views

Author