A360005 -id:A360005 - cp's OEIS frontend

A359912 Numbers whose prime indices do not have integer median.

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 60, 65, 69, 74, 77, 84, 86, 93, 95, 106, 119, 122, 123, 132, 141, 142, 143, 145, 150, 156, 158, 161, 177, 178, 185, 196, 201, 202, 204, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 276, 278

Offset: 1

Gus Wiseman, Jan 24 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 terms together with their prime indices begin:
   1: {}
   6: {1,2}
  14: {1,4}
  15: {2,3}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  51: {2,7}
  58: {1,10}
  60: {1,1,2,3}

For prime factors instead of indices we have A072978, complement A359913.
These partitions are counted by A307683.
For mean instead of median: A348551, complement A316413, counted by A349156.
The complement is A359908, counted by A325347.
Positions of odd terms in A360005.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A051293, A067538, A175352, A175761, A289509, A359890, A359905, A360006, A359907, A360009.

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

A359890 Numbers whose prime indices do not have the same mean as median.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 92, 96, 98, 99, 102, 104, 108, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160, 162, 164, 165

Offset: 1

Gus Wiseman, Jan 22 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 terms together with their prime indices begin:
   12: {1,1,2}
   18: {1,2,2}
   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}
   45: {2,2,3}
   48: {1,1,1,1,2}
For example, the prime indices of 360 are {1,1,1,2,2,3}, with mean 5/3 and median 3/2, so 360 is in the sequence.

The LHS (mean of prime indices) is A326567/A326568.
The complement is A359889, counted by A240219.
The odd-length case is A359891, complement A359892.
These partitions are counted by A359894.
The strict case is counted by A359898, odd-length A359900.
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.
A359908 lists numbers whose prime indices have integer median.
Cf. A327473, A327476, A348551, A359903, A359911, A359912, A360006-A360009.

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

A360006 Least positive integer whose prime indices have median n/2. a(1) = 1.

Original entry on oeis.org

1, 2, 6, 3, 14, 5, 26, 7, 38, 11, 58, 13, 74, 17, 86, 19, 106, 23, 122, 29, 142, 31, 158, 37, 178, 41, 202, 43, 214, 47, 226, 53, 262, 59, 278, 61, 302, 67, 326, 71, 346, 73, 362, 79, 386, 83, 398, 89, 446, 97, 458, 101, 478, 103, 502, 107, 526, 109, 542, 113

Offset: 1

Gus Wiseman, Jan 24 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).

Position of first appearance of n in A360005.
The sorted version is A360007, for mean A360008.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 counts partitions by median, cf. A359901, A359902.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A359889, A359890, A360009.

nn=100;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,2*Median[prix[n]]],{n,nn}];
Table[Position[seq,k][[1,1]],{k,Count[Differences[Union[seq]],1]}]

Consists of 1 followed by A000040 interleaved with 2*A031215.

A360672 Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 0, 0, 3, 6, 4, 1, 0, 1, 0, 0, 1, 7, 7, 5, 1, 0, 1, 0, 0, 1, 4, 8, 10, 5, 1, 0, 1, 0, 0, 0, 3, 6, 14, 11, 6, 1, 0, 1, 0, 0, 0, 1, 5, 12, 16, 14, 6, 1, 0

Offset: 0

Gus Wiseman, Feb 27 2023

Also the number of integer partitions of n whose right half (inclusive) sums to n-k.

			Triangle begins:
  1
  1  0
  1  1  0
  1  1  1  0
  1  0  3  1  0
  1  0  2  3  1  0
  1  0  1  4  4  1  0
  1  0  0  3  6  4  1  0
  1  0  0  1  7  7  5  1  0
  1  0  0  1  4  8 10  5  1  0
  1  0  0  0  3  6 14 11  6  1  0
  1  0  0  0  1  5 12 16 14  6  1  0
  1  0  0  0  1  2 12 14 23 16  7  1  0
  1  0  0  0  0  2  7 13 24 27 19  7  1  0
  1  0  0  0  0  1  5  9 24 30 35 21  8  1  0
  1  0  0  0  0  1  3  7 17 31 42 40 25  8  1  0
  1  0  0  0  0  0  2  4 16 23 46 51 51 27  9  1  0
  1  0  0  0  0  0  1  3 10 21 37 57 69 57 31  9  1  0
  1  0  0  0  0  0  1  2  7 15 34 47 83 81 69 34 10  1  0
For example, row n = 9 counts the following partitions:
  (9)  .  .  (333)  (432)        (54)        (63)      (72)    (81)
                    (441)        (522)       (621)     (711)
                    (22221)      (531)       (3321)    (4311)
                    (111111111)  (3222)      (4221)    (5211)
                                 (32211)     (33111)   (6111)
                                 (2211111)   (42111)
                                 (3111111)   (51111)
                                 (21111111)  (222111)
                                             (321111)
                                             (411111)
For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).

Row sums are A000041.
Column sums are A360673, inclusive A360671.
The central diagonal T(2n,n) is A360674, ranks A360953.
The left inclusive version is A360675 with rows reversed.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616, A360682, A360686.

Table[Length[Select[IntegerPartitions[n], Total[Take[#,Floor[Length[#]/2]]]==k&]],{n,0,10},{k,0,n}]

A360675 Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 3, 3, 0, 0, 0, 1, 3, 5, 2, 0, 0, 0, 1, 4, 6, 4, 0, 0, 0, 0, 1, 4, 9, 5, 3, 0, 0, 0, 0, 1, 5, 10, 10, 4, 0, 0, 0, 0, 0, 1, 5, 13, 12, 9, 2, 0, 0, 0, 0, 0, 1, 6, 15, 18, 11, 5, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 27 2023

Also the number of integer partitions of n whose left half (inclusive) sums to n-k.

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  2  2  0  0
  1  3  3  0  0  0
  1  3  5  2  0  0  0
  1  4  6  4  0  0  0  0
  1  4  9  5  3  0  0  0  0
  1  5 10 10  4  0  0  0  0  0
  1  5 13 12  9  2  0  0  0  0  0
  1  6 15 18 11  5  0  0  0  0  0  0
  1  6 18 22 20  6  4  0  0  0  0  0  0
  1  7 20 29 26 13  5  0  0  0  0  0  0  0
  1  7 24 34 37 19 11  2  0  0  0  0  0  0  0
  1  8 26 44 46 30 16  5  0  0  0  0  0  0  0  0
  1  8 30 50 63 40 27  8  4  0  0  0  0  0  0  0  0
  1  9 33 61 75 61 36 15  6  0  0  0  0  0  0  0  0  0
  1  9 37 70 96 75 61 21 12  3  0  0  0  0  0  0  0  0  0
For example, row n = 9 counts the following partitions:
  (9)  (81)   (72)     (63)       (54)
       (441)  (432)    (333)      (3222)
       (531)  (522)    (3321)     (21111111)
       (621)  (4311)   (4221)     (111111111)
       (711)  (5211)   (22221)
              (6111)   (222111)
              (32211)  (321111)
              (33111)  (411111)
              (42111)  (2211111)
              (51111)  (3111111)
For example, the partition y = (3,2,2,1,1) has right half (exclusive) (1,1), with sum 2, so y is counted under T(9,2).

The central diagonal T(2n,n) is A000005.
Row sums are A000041.
Diagonal sums are A360671, exclusive A360673.
The right inclusive version is A360672 with rows reversed.
The left version has central diagonal A360674, ranks A360953.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616.

Table[Length[Select[IntegerPartitions[n], Total[Take[#,-Floor[Length[#]/2]]]==k&]],{n,0,18},{k,0,n}]

A124943 Table read by rows: number of partitions of n with k as low median.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1, 6, 3, 1, 0, 0, 1, 8, 4, 2, 0, 0, 0, 1, 11, 6, 3, 1, 0, 0, 0, 1, 15, 8, 4, 2, 0, 0, 0, 0, 1, 20, 12, 5, 3, 1, 0, 0, 0, 0, 1, 26, 16, 7, 4, 2, 0, 0, 0, 0, 0, 1, 35, 22, 10, 5, 3, 1, 0, 0, 0, 0, 0, 1, 45, 29, 14, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 58, 40, 19, 8, 5, 3, 1

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the low median is the same as the median. For a multiset with an even number of elements, the low median is the smaller of the two central elements.

Arrange the parts of a partition nonincreasing order. Remove the first part, then the last, then the first remaining part, then the last remaining part, and continue until only a single number, the low median, remains. - Clark Kimberling, May 16 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the low median. For [3,2,1^2], the two middle elements are 1 and 2; the low median is the smaller, 1.
First 8 rows:
  1
  1   1
  2   0   1
  3   1   0   1
  4   2   0   0   1
  6   3   1   0   0   1
  8   4   2   0   0   0   1
  11  6   3   1   0   0   0   1
From _Gus Wiseman_, Jul 09 2023: (Start)
Row n = 8 counts the following partitions:
  (71)        (62)     (53)   (44)  .  .  .  (8)
  (611)       (521)    (431)
  (5111)      (422)    (332)
  (4211)      (3221)
  (41111)     (2222)
  (3311)      (22211)
  (32111)
  (311111)
  (221111)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336, ranks A363488.
The high version of this triangle is A124944.
The rank statistic for this triangle is A363941, high version A363942.
A version for mean instead of median is A363945, rank statistic A363943.
A high version for mean instead of median is A363946, rank stat A363944.
A version for mode instead of median is A363952, high A363953.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A025065, A026794, A027193, A067538, A237984, A240219, A362608, A363740.

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 2)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Jun 23 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.
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 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:
     2: {1}          29: {10}              79: {22}
     3: {2}          31: {11}              81: {2,2,2,2}
     4: {1,1}        32: {1,1,1,1,1}       83: {23}
     5: {3}          37: {12}              89: {24}
     7: {4}          41: {13}              90: {1,2,2,3}
     8: {1,1,1}      43: {14}              97: {25}
     9: {2,2}        47: {15}             101: {26}
    11: {5}          49: {4,4}            103: {27}
    13: {6}          53: {16}             107: {28}
    16: {1,1,1,1}    59: {17}             109: {29}
    17: {7}          61: {18}             113: {30}
    19: {8}          64: {1,1,1,1,1,1}    121: {5,5}
    23: {9}          67: {19}             125: {3,3,3}
    25: {3,3}        71: {20}             127: {31}
    27: {2,2,2}      73: {21}             128: {1,1,1,1,1,1,1}

These partitions are counted by A363719, factorizations A363741.
For unequal instead of equal we have A363730, counted by A363720.
Excluding primes gives A363722.
Excluding prime-powers gives A363729, counted by A363728.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

Assuming there is a unique mode, we have A326567(a(n))/A326568(a(n)) = A360005(a(n))/2 = A363486(a(n)) = A363487(a(n)).

A360248 Numbers for which the prime indices do not have the same median as the distinct prime indices.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 200

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

First differs from A242416 in lacking 180, with prime indices {1,1,2,2,3}.
First differs from A360246 in lacking 126 and having 1950.
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 terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is not in the sequence.
The prime indices of 1950 are {1,2,3,3,6} with median 3 and distinct prime indices {1,2,3,6} with median 5/2, so 1950 is in the sequence.

These partitions are counted by A360244.
The complement is A360249, counted by A360245.
For multiplicities instead of parts: complement of A360453.
For multiplicities instead of distinct parts: complement of A360454.
For mean instead of median we have A360246, counted by A360242.
The complement for mean instead of median is A360247, counted by A360243.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A325347 = partitions with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median.
A360005 gives median of prime indices (times two).
Cf. A000975, A078174, A316413, A324570, A359890, A360455, A360456.

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

A361849 Number of integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 3, 4, 7, 9, 9, 15, 16, 20, 26, 34, 37, 50, 55, 68, 86, 103, 117, 145, 168, 201, 236, 282, 324, 391, 449, 525, 612, 712, 818, 962, 1106, 1278, 1470, 1698, 1939, 2238, 2550, 2924, 3343, 3824, 4341, 4963, 5627, 6399, 7256, 8231, 9300

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) = 9 partitions:
  211  2111  21111  421     422      4221      631        632
                    3211    221111   4311      4222       5321
                    22111   2111111  2211111   42211      5411
                    211111           21111111  322111     42221
                                               2221111    43211
                                               22111111   332111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
For example, the partition (3,2,1,1) has maximum 3 and median 3/2, so is counted under a(7).

For minimum instead of median we have A118096.
For length instead of median we have A237753.
This is the equal case of A361848.
For mean instead of median we have A361853.
These partitions have ranks A361856.
For "greater" instead of "equal" we have A361857, allowing equality A361859.
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.
A361860 counts partitions with minimum equal to median.
Cf. A013580, A027193, A053263, A061395, A067659, A111907, A116608, A237755, A240219, A361851.

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

A124944 Table, number of partitions of n with k as high median.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017
Arrange the parts of a partition nonincreasing order.  Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
From _Gus Wiseman_, Jul 12 2023: (Start)
Triangle begins:
   1
   1  1
   1  1  1
   2  1  1  1
   3  1  1  1  1
   4  3  1  1  1  1
   6  4  1  1  1  1  1
   8  6  3  1  1  1  1  1
  11  8  5  1  1  1  1  1  1
  15 11  7  3  1  1  1  1  1  1
  20 15  9  5  1  1  1  1  1  1  1
  26 21 12  8  3  1  1  1  1  1  1  1
  35 27 16 10  5  1  1  1  1  1  1  1  1
  45 37 21 13  8  3  1  1  1  1  1  1  1  1
  58 48 29 16 11  5  1  1  1  1  1  1  1  1  1
Row n = 8 counts the following partitions:
  (611)       (521)    (431)   (44)  (53)  (62)  (71)  (8)
  (5111)      (422)    (332)
  (41111)     (4211)   (3311)
  (32111)     (3221)
  (311111)    (2222)
  (221111)    (22211)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336(n-1), ranks A364056.
Column k = 1 in the low version is A027336, ranks A363488.
The low version of this triangle is A124943.
The rank statistic for this triangle is A363942, low version A363941.
A version for mean instead of median is A363946, low A363945.
A version for mode instead of median is A363953, low A363952.
A008284 counts partitions by length, maximum, or decreasing mean.
A026794 counts partitions by minimum, strict A026821.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A008289, A025065, A027193, A067538, A237984, A240219, A362608, A363740, A363943, A363944.

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

cp's OEIS Frontend

A359912 Numbers whose prime indices do not have integer median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359890 Numbers whose prime indices do not have the same mean as median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360006 Least positive integer whose prime indices have median n/2. a(1) = 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A360672 Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360675 Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A124943 Table read by rows: number of partitions of n with k as low median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A360248 Numbers for which the prime indices do not have the same median as the distinct prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A361849 Number of integer partitions of n such that the maximum is twice the median.

Views

Author

Keywords

Comments

Examples