A359907 -id:A359907 - cp's OEIS frontend

A361911 Number of set partitions of {1..n} with block-medians summing to an integer.

1, 1, 3, 10, 30, 107, 479, 2249, 11173, 60144, 351086, 2171087, 14138253, 97097101, 701820663, 5303701310, 41838047938, 343716647215, 2935346815495, 25999729551523, 238473713427285, 2261375071834708, 22141326012712122, 223519686318676559, 2323959300370456901

Offset: 1

Gus Wiseman, Apr 14 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(4) = 10 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with sum 5, so is counted under a(4).

For median instead of sum we have A361864.
For mean of means we have A361865.
For mean instead of median we have A361866.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A308037 counts set partitions with integer average block-size.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A007837, A035470, A038041, A079309, A231147, A275714, A275780, A359893, A361801, A361910.

sps[{}]:={{}}; sps[set:{i_,_}] := Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], IntegerQ[Total[Median/@#]]&]],{n,10}]

a(12)-a(25) from Christian Sievers, Aug 26 2024

A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166, 177

Offset: 1

Gus Wiseman, May 10 2023

nonn

Also numbers n whose median prime factor is not a prime factor of n, where 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 factorization of 60 is 2*2*3*5, with middle parts (2,3), so 60 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Partitions of this type are counted by A238479.
The complement (without 1) is A362618, counted by A238478.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A359893 counts partitions by median.
A359908 ranks partitions with integer median, counted by A325347.
A359912 ranks partitions with non-integer median, counted by A307683.
A362605 ranks partitions with more than one mode, counted by A362607.
A362611 counts modes in prime factorization, triangle version A362614.
A362621 ranks partitions with median equal to maximum, counted by A053263.
A362622 ranks partitions whose maximum is a middle part, counted by A237824.
Contains A006881 and (except for 1) A030229.
Cf. A000040, A171979, A327473, A327476, A356862, A359907, A362616, A362620.

filter:= proc(n) local F,m;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= nops(F);
  m::even and F[m/2] <> F[m/2+1]
end proc:
select(filter, [$2..200]); # Robert Israel, Dec 15 2023

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[prifacs[#],Median[prifacs[#]]]&]

A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 2, 5, 7, 10, 14, 21, 28, 36, 51, 64, 84, 106, 132, 165, 202, 252, 311, 391, 473, 579, 713, 868, 1069, 1303, 1617, 1954, 2404, 2908, 3556, 4282, 5200, 6207, 7505, 8934, 10700, 12717, 15165, 17863, 21222, 24976, 29443, 34523, 40582, 47415

Offset: 0

Gus Wiseman, Feb 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(11) = 10 partitions:
  1   .  .  22   .  .  2221   3311    333      4222      5222
                              32111   3222     33211     33221
                                      32211    42211     52211
                                      42111    43111     53111
                                      321111   52111     62111
                                               421111    322211
                                               3211111   431111
                                                         521111
                                                         4211111
                                                         32111111

For mean instead of median: A360068, ranks A359903.
For distinct parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360454.
For distinct parts instead of parts: A360455, ranks A360453.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A114638, A240219, A359894, A359895, A360244, A360248.

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

A362618 Numbers whose prime factorization has either (1) odd length, or (2) equal middle parts.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 90, 92, 96, 97, 98, 99, 101

Offset: 1

Gus Wiseman, May 10 2023

nonn

Also numbers n whose median prime factor is a prime factor of n.

			The prime factorization of 90 is 2*3*3*5, with middle parts (3,3), so 90 is in the sequence.

Partitions of this type are counted by A238478.
The complement (without 1) is A362617, counted by A238479.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A359893 counts partitions by median.
A359908 ranks partitions with integer median, counted by A325347.
A359912 ranks partitions with non-integer median, counted by A307683.
A362611 ranks modes in prime factorization, counted by A362614.
A362621 ranks partitions with median equal to maximum, counted by A053263.
A362622 ranks partitions whose maximum is a middle part, counted by A237824.
Cf. A000040, A171979, A327473, A327476, A356862, A359907, A360686, A360952, A362616, A362620.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],MemberQ[prifacs[#],Median[prifacs[#]]]&]

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

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 terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
   30: {1,2,3}
   42: {1,2,4}
   49: {4,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.

For distinct prime indices instead of 0-prepended differences: A360453.
For mean instead of median we have A360680.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Multisets with integer median:
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#],0]]]&]

A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899

Offset: 0

Gus Wiseman, Feb 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).

			The a(3) = 1 through a(9) = 13 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (321)     (31111)    (3221)      (432)
                          (2211)    (211111)   (3311)      (531)
                          (21111)   (1111111)  (22211)     (22221)
                          (111111)             (32111)     (33111)
                                               (41111)     (51111)
                                               (221111)    (222111)
                                               (311111)    (321111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

For first differences we have A237363.
For sum instead of median we have A360683.
For mean instead of median we have A360683 - A008619.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives median of prime indices (times two).
Cf. A000975, A027193, A067538, A114638, A307683, A359908, A360245, A360254, A360687, A360688.

Table[Length[Select[IntegerPartitions[n],Median[Differences[#,2]]==0&]],{n,0,30}]

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

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(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).

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.

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

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Gus Wiseman, Feb 22 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) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A360691 Number of integer partitions of n with non-integer median of 0-prepended first differences.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213

Offset: 1

Gus Wiseman, Feb 22 2023

nonn

All of these partitions have even length.

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) = 0 through a(10) = 10 partitions:
  .  (11)  .  (31)  (32)    (33)    (52)    (53)    (54)      (55)
                    (2111)  (51)    (2221)  (71)    (72)      (73)
                            (2211)  (4111)  (3311)  (3222)    (91)
                            (3111)          (5111)  (6111)    (3322)
                                                    (321111)  (3331)
                                                              (4411)
                                                              (5311)
                                                              (7111)
                                                              (322111)
                                                              (421111)

For median 0 we have A360254, ranks A360558.
These partitions have ranks A360557, complement A360556.
The complement is counted by A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A237363, A240219, A359907, A359908, A360455, A360555.

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]

A361391 Number of strict integer partitions of n with non-integer mean.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 4, 2, 4, 5, 11, 0, 17, 15, 13, 15, 37, 18, 53, 24, 48, 78, 103, 23, 111, 152, 143, 123, 255, 110, 339, 238, 372, 495, 377, 243, 759, 845, 873, 414, 1259, 842, 1609, 1383, 1225, 2281, 2589, 1285, 2827, 2518, 3904, 3836, 5119, 3715, 4630

Offset: 0

Gus Wiseman, Mar 11 2023

nonn

Are 1, 2, 4, 6, 12 the only zeros?

			The a(3) = 1 through a(11) = 11 partitions:
  {2,1}  .  {3,2}  .  {4,3}    {4,3,1}  {5,4}  {5,3,2}    {6,5}
            {4,1}     {5,2}    {5,2,1}  {6,3}  {5,4,1}    {7,4}
                      {6,1}             {7,2}  {6,3,1}    {8,3}
                      {4,2,1}           {8,1}  {7,2,1}    {9,2}
                                               {4,3,2,1}  {10,1}
                                                          {5,4,2}
                                                          {6,3,2}
                                                          {6,4,1}
                                                          {7,3,1}
                                                          {8,2,1}
                                                          {5,3,2,1}

The strict complement is counted by A102627.
The non-strict version is ranked by A348551, complement A316413.
The non-strict version is counted by A349156, complement A067538.
For median instead of mean we have A360952, complement A359907.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A307683 counts partitions with non-integer median, ranks A359912.
A325347 counts partitions with integer median, ranks A359908.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement of A237984.
A327475 counts subsets with integer mean.
Cf. A051293, A082550, A143773, A175397, A175761, A240219, A240850, A326027, A326641, A326849, A359897.

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Mar 16 2023

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

a(31)-a(55) from Alois P. Heinz, Mar 16 2023

cp's OEIS Frontend

A361911 Number of set partitions of {1..n} with block-medians summing to an integer.

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A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

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A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

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A362618 Numbers whose prime factorization has either (1) odd length, or (2) equal middle parts.

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A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

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A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

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

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A360690 Number of integer partitions of n with non-integer median of multiplicities.

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Mathematica

A360691 Number of integer partitions of n with non-integer median of 0-prepended first differences.

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