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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A361910 Number of set partitions of {1..n} such that the mean of the means of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 7, 12, 47, 99, 430, 1379, 5613, 21416, 127303, 532201, 3133846, 18776715, 114275757, 737859014
Offset: 1

Views

Author

Gus Wiseman, Apr 14 2023

Keywords

Comments

Since (n+1)/2 is the mean of {1..n}, this sequence counts a type of "transitive" set partitions.

Examples

			The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}      {{1245}{3}}
                   {{1}{2}{3}}  {{13}{24}}      {{135}{24}}
                                {{14}{23}}      {{15}{234}}
                                {{1}{23}{4}}    {{1}{234}{5}}
                                {{14}{2}{3}}    {{12}{3}{45}}
                                {{1}{2}{3}{4}}  {{135}{2}{4}}
                                                {{14}{25}{3}}
                                                {{15}{24}{3}}
                                                {{1}{24}{3}{5}}
                                                {{15}{2}{3}{4}}
                                                {{1}{2}{3}{4}{5}}
The set partition {{1,3},{2,4}} has means {2,3}, with mean 5/2, so is counted under a(4).
The set partition {{1,3,5},{2,4}} has means {3,3}, with mean 3, so is counted under a(5).

Crossrefs

For median instead of mean we have A361863.
A000110 counts set partitions.
A308037 counts set partitions with integer mean block-size.
A327475 counts subsets with integer mean, A000975 with integer median.
A327481 counts subsets by mean, A013580 by median.
A361865 counts set partitions with integer mean of means.
A361911 counts set partitions with integer sum of means.
Cf. A007837, A035470, A038041, A067538, A275714, A275780, A326512, A326513, A361864, A361866.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Mean[Join@@#]==Mean[Mean/@#]&]],{n,8}]

Extensions

a(13)-a(17) from Christian Sievers, May 12 2025

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}]

A363223 Numbers with bigomega equal to median prime index.

Original entry on oeis.org

2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970
Offset: 1

Views

Author

Gus Wiseman, May 29 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).
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:
    2: {1}
    9: {2,2}
   10: {1,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  130: {1,3,6}
  165: {2,3,5}
  170: {1,3,7}
  175: {3,3,4}

Crossrefs

For maximum instead of median we have A106529, counted by A047993.
For minimum instead of median we have A324522, counted by A006141.
Partitions of this type are counted by A361800.
For twice median we have A362050, counted by A362049.
For maximum instead of length we have A362621, counted by A053263.
A000975 counts subsets with integer median.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A359908 lists numbers whose prime indices have integer median.
A360005 gives twice median of prime indices.
Cf. A000040, A013580, A079309, A240219, A327473, A327476, A361860, A362619, A362622, A362980.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==Median[prix[#]]&]

Formula

2*A001222(a(n)) = A360005(a(n)).

A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 9, 26, 69, 335, 1018, 6629, 22805, 182988, 703745
Offset: 1

Views

Author

Gus Wiseman, Apr 04 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).
Since (n+1)/2 is the median of {1..n}, this sequence counts "transitive" set partitions.

Examples

			The a(1) = 1 through a(4) = 9 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{23}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with median 5/2, so is counted under a(4).
The set partition {{1,3},{2,4}} has medians {2,3}, with median 5/2, so is counted under a(4).

Crossrefs

For mean instead of median we have A361910.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A325347 counts partitions w/ integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361864 counts set partitions with integer median of medians, means A361865.
A361866 counts set partitions with integer sum of medians, means A361911.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A326516, A326521, A326537, A359907, A361654, A361801.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],(n+1)/2==Median[Median/@#]&]],{n,6}]

A361802 Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 5, 2, 1, 1, 4, 10, 16, 18, 14, 8, 3, 1, 1, 5, 15, 31, 46, 51, 43, 27, 12, 3, 1, 1, 6, 21, 53, 98, 139, 155, 134, 88, 43, 16, 4, 1, 1, 7, 28, 83, 184, 319, 441, 486, 424, 293, 161, 68, 21, 4, 1
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2023

Keywords

Comments

Also the number of k-subsets of {1,...,2n} with mean n.

Examples

			Triangle begins:
   1
   1   1   1
   1   2   3   2   1
   1   3   6   7   5   2   1
   1   4  10  16  18  14   8   3   1
   1   5  15  31  46  51  43  27  12   3   1
   1   6  21  53  98 139 155 134  88  43  16   4   1
   1   7  28  83 184 319 441 486 424 293 161  68  21   4   1
Row n = 3 counts the following subsets:
  {0}  {-1,1}  {-1,0,1}   {-2,-1,0,3}  {-2,-1,0,1,2}
       {-2,2}  {-2,0,2}   {-2,-1,1,2}
               {-2,-1,3}

Crossrefs

Row lengths are A005408.
Row sums are A212352.
A007318 counts subsets by length.
A067538 counts partitions with integer mean.
A231147 counts subsets by median.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean.
Cf. A006134, A013580, A024718, A079309, A326512, A349156, A361654, A362046.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[-n+1,n],{k}],Total[#]==0&]],{n,6},{k,2n-1}]
Previous Showing 21-25 of 25 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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