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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A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 10, 15, 23, 34, 46, 67, 90, 121, 164, 219, 285, 375, 483, 622, 799, 1017, 1284, 1621, 2033, 2537, 3158, 3915, 4832, 5953, 7303, 8930, 10896, 13248, 16071, 19451, 23482, 28272, 33977, 40736, 48741, 58201, 69367, 82506, 97986, 116139
Offset: 1

Views

Author

Gus Wiseman, Apr 02 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(4) = 1 through a(9) = 15 partitions:
  (211)  (311)   (411)    (421)     (422)      (522)
         (2111)  (3111)   (511)     (521)      (621)
                 (21111)  (3211)    (611)      (711)
                          (4111)    (4211)     (4221)
                          (22111)   (5111)     (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (33111)
                                    (311111)   (42111)
                                    (2111111)  (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).

Crossrefs

For length instead of median we have A237752.
For minimum instead of median we have A237821.
Reversing the inequality gives A361848.
The equal case is A361849, ranks A361856.
The unequal case is A361857, ranks A361867.
The complement is counted by A361858.
These partitions have ranks A361868.
For mean instead of median we have A361906.
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, A027193, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

Programs

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

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

First differs from A067340 in having 60.
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).

Examples

			The unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

Crossrefs

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- 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.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360687 Number of integer partitions of n whose multiplicities have integer median.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676
Offset: 1

Views

Author

Gus Wiseman, Feb 20 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(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
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 counted under a(8).

Crossrefs

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
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.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

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

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103
Offset: 1

Views

Author

Gus Wiseman, Feb 16 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 prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

Crossrefs

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- 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.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

A360686 Number of integer partitions of n whose distinct parts have integer median.

Original entry on oeis.org

1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408
Offset: 1

Views

Author

Gus Wiseman, Feb 20 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(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (421)      (53)
                    (1111)           (51)      (511)      (62)
                                     (222)     (3211)     (71)
                                     (321)     (31111)    (422)
                                     (3111)    (1111111)  (431)
                                     (111111)             (521)
                                                          (2222)
                                                          (3221)
                                                          (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).

Crossrefs

For all parts: A325347, strict A359907, ranks A359908, complement A307683.
For mean instead of median: A360241, ranks A326621.
These partitions have ranks A360550, complement A360551.
For multiplicities instead of distinct parts: A360687.
The complement is counted by A360689.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A027193 counts odd-length partitions, strict A067659, ranks A026424.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A240219, A359906, A360005, A360071, A360244, A360245, A360556, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Union[#]]]&]],{n,30}]

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Apr 02 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(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).

Crossrefs

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.

Programs

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

A361864 Number of set partitions of {1..n} whose block-medians have integer median.

Original entry on oeis.org

1, 0, 3, 6, 30, 96, 461, 2000, 10727, 57092, 342348
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).

Examples

			The a(1) = 1 through a(4) = 6 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}
            {{13}{2}}    {{123}{4}}
            {{1}{2}{3}}  {{1}{2}{34}}
                         {{12}{3}{4}}
                         {{1}{24}{3}}
                         {{13}{2}{4}}
The set partition {{1,2},{3},{4}} has block-medians {3/2,3,4}, with median 3, so is counted under a(4).

Crossrefs

For mean instead of median we have A361865.
For sum instead of outer median we have A361911, means 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 counts partitions w/ integer median, complement A307683.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A359893, A359907, 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]],IntegerQ[Median[Median/@#]]&]],{n,6}]

A360952 Number of strict integer partitions of n with non-integer median; a(0) = 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 6, 1, 8, 4, 11, 5, 15, 10, 20, 13, 27, 22, 36, 28, 47, 43, 63, 56, 82, 79, 107, 103, 140, 141, 180, 181, 232, 242, 299, 308, 380, 402, 483, 511, 613, 656, 772, 824, 969, 1047, 1215, 1309, 1514, 1642, 1882, 2039, 2334, 2539, 2882
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2023

Keywords

Comments

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).

Examples

			The a(0) = 1 through a(15) = 11 partitions (0 = {}, A..E = 10..14):
  0  .  .  21  .  32  .  43  .  54  4321  65    6321  76    5432  87
                  41     52     63        74          85    6431  96
                         61     72        83          94    6521  A5
                                81        92          A3    8321  B4
                                          A1          B2          C3
                                          5321        C1          D2
                                                      5431        E1
                                                      7321        6432
                                                                  7431
                                                                  7521
                                                                  9321

Crossrefs

The non-strict version is A307683, ranks A359912.
The non-strict complement is A325347, ranks A359908.
The strict complement is counted by A359907.
For mean instead of median we have A361391, non-strict A349156.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 = partitions with integer mean, complement A102627, ranks A316413.
A359893/A359901/A359902 count partitions by median.
A360005(n)/2 ranks the median statistic.
Cf. A000975, A051293, A082550, A240219, A240850, A327475, A359897.

Programs

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

Formula

a(n) = A000009(n) - A359907(n).

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

Views

Author

Gus Wiseman, Apr 05 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.
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 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}

Crossrefs

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.

Programs

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

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Apr 14 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(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).

Crossrefs

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.

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]], IntegerQ[Total[Median/@#]]&]],{n,10}]

Extensions

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