A363488 -id:A363488 - cp's OEIS frontend

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

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 *)

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 *)

A363941 Low median in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 2, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 2, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 2, 14, 1, 2, 1, 15, 1, 4, 3, 2, 1, 16, 2, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 2, 19, 1, 2, 3, 20, 1, 21, 1, 3, 1, 4, 2, 22, 1, 2, 1

Offset: 1

Gus Wiseman, Jul 01 2023

nonn

The low median (see A124943) in a multiset is either the middle part (for odd length), or the least 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.

			The prime indices of 90 are {1,2,2,3}, with low median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with low median 2, so a(150) = 2.

Positions of first appearances are 1 and A000040.
The triangle for this statistic (low median) is A124943, high A124944.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A363486, high A363487.
Positions of 1's are A363488.
The high version is A363942.
A067538 counts partitions with integer mean, ranked by A316413.
A112798 lists prime indices, length A001222, sum A056239.
A363943 gives low mean of prime indices, triangle A363945.
A363944 gives high mean of prime indices, triangle A363946.
Cf. A025065, A215366, A326567/A326568, A344296, A359889, A359908, A363727, A363740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mell[y_]:=If[Length[y]==0,0, If[OddQ[Length[y]],y[[(Length[y]+1)/2]],y[[Length[y]/2]]]];
Table[mell[prix[n]],{n,30}]

A363949 Numbers whose prime indices have mean 1 when rounded down.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 560, 576, 600, 640

Offset: 1

Gus Wiseman, Jul 02 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 terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

These partitions are counted by A025065.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A360015, counted by A241131.
For median instead of mean we have A363488, counted by A027336.
Positions of 1's in A363943, triangle A363945.
For the usual rounding (not low or high) we have A363948, counted by A363947.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
For mean 2 instead of 1 we have A363950, counted by A026905 redoubled.
Cf. A051293, A327473, A327476, A344296, A359889, A360013, A363723, A363727, A363946, A363951.

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

a(n) = 2*A344296(n).

A364158 Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 18, 22, 26, 30, 32, 34, 36, 38, 42, 46, 50, 54, 58, 62, 64, 66, 70, 74, 78, 82, 86, 90, 94, 98, 100, 102, 106, 108, 110, 114, 118, 122, 126, 128, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194

Offset: 1

Gus Wiseman, Jul 14 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.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Except for 1, this is the lists of all even numbers whose prime factorization contains at most as many 2's as non-2 parts.
Extending the terminology of A124943, the "low co-mode" of a multiset is the least co-mode.

			The terms together with their prime factorizations begin:
   1 =
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  10 = 2*5
  14 = 2*7
  16 = 2*2*2*2
  18 = 2*3*3
  22 = 2*11
  26 = 2*13
  30 = 2*3*5
  32 = 2*2*2*2*2
  34 = 2*17
  36 = 2*2*3*3

Partitions of this type are counted by A364159.
Positions of 1's in A364191, high A364192, modes A363486, high A363487.
For median we have A363488, positions of 1 in A363941, triangle A124943.
For mode instead of co-mode we have A360015, counted by A241131.
A027746 lists prime factors (with multiplicity), length A001222.
A362611 counts modes in prime factorization, triangle A362614
A362613 counts co-modes in prime factorization, triangle A362615
Ranking partitions:
- A356862: unique mode, counted by A362608
- A359178: unique co-mode, counted by A362610
- A362605: multiple modes, counted by A362607
- A362606: multiple co-modes, counted by A362609
Cf. A215366, A327473, A327476, A360005.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Select[Range[100],#==1||Min[comodes[prifacs[#]]]==2&]

A364191 Low co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 1, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 1, 2, 6, 16, 1, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 2, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Jul 16 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.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "low co-mode" in a multiset is the least co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 2.

For prime factors instead of indices we have A067695, high A359612.
For mode instead of co-mode we have A363486, high A363487, triangle A363952.
For median instead of co-mode we have A363941, high A363942.
Positions of 1's are A364158, counted by A364159.
The high version is A364192 = positions of 1's in A364061.
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A124943, A241131, A327473, A327476, A360005, A360015, A363488.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Min[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A067695(n)).

A067695(n) = A000040(a(n)).

A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 104, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 264, 268, 272, 280, 284, 288, 292

Offset: 1

Gus Wiseman, Jul 07 2023

nonn

The multiset of prime factors of n is row n of A027746.

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).

			The terms together with their prime indices begin:
     2: {1}             64: {1,1,1,1,1,1}      136: {1,1,1,7}
     4: {1,1}           68: {1,1,7}            144: {1,1,1,1,2,2}
     8: {1,1,1}         72: {1,1,1,2,2}        148: {1,1,12}
    12: {1,1,2}         76: {1,1,8}            152: {1,1,1,8}
    16: {1,1,1,1}       80: {1,1,1,1,3}        160: {1,1,1,1,1,3}
    20: {1,1,3}         88: {1,1,1,5}          164: {1,1,13}
    24: {1,1,1,2}       92: {1,1,9}            168: {1,1,1,2,4}
    28: {1,1,4}         96: {1,1,1,1,1,2}      172: {1,1,14}
    32: {1,1,1,1,1}    104: {1,1,1,6}          176: {1,1,1,1,5}
    40: {1,1,1,3}      112: {1,1,1,1,4}        184: {1,1,1,9}
    44: {1,1,5}        116: {1,1,10}           188: {1,1,15}
    48: {1,1,1,1,2}    120: {1,1,1,2,3}        192: {1,1,1,1,1,1,2}
    52: {1,1,6}        124: {1,1,11}           200: {1,1,1,3,3}
    56: {1,1,1,4}      128: {1,1,1,1,1,1,1}    208: {1,1,1,1,6}

Partitions of this type are counted by A027336.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A360013, low A360015.
The low version is A363488, positions of 1's in A363941.
Positions of 1's in A363942.
A112798 lists prime indices, length A001222, sum A056239.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A124943 counts partitions by low median, high A124944.
Cf. A072978, A215366, A316413, A359908, A363727, A363740, A363949.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
merr[y_]:=If[Length[y]==0,0,If[OddQ[Length[y]],y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];
Select[Range[100],merr[prifacs[#]]==2&]

A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 66, 90, 118, 156, 201, 264, 336, 434, 550, 700, 880, 1112, 1385, 1733, 2149, 2666, 3283, 4049, 4956, 6072, 7398, 9009, 10922, 13237, 15970, 19261, 23147, 27790, 33260, 39776, 47425, 56497, 67133, 79685, 94371, 111653

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - Gus Wiseman, Jul 10 2023

			a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
From _Gus Wiseman_, Jul 10 2023 (Start)
The a(2) = 1 through a(8) = 14 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)
       (21)  (22)  (32)   (33)    (43)    (44)
             (31)  (41)   (42)    (52)    (53)
                   (221)  (51)    (61)    (62)
                          (222)   (322)   (71)
                          (321)   (331)   (332)
                          (2211)  (421)   (422)
                                  (2221)  (431)
                                  (3211)  (521)
                                          (2222)
                                          (3221)
                                          (3311)
                                          (4211)
                                          (22211)
(End)

Cf. A096373.
For mean instead of median we have A000065, ranks A057716.
The complement is counted by A027336, ranks A364056.
Rows sums of A359893 if we remove the first column.
These partitions have ranks A364058.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A124943 counts partitions by low median, high A124944.
A241131 counts partitions with low mode 1, ranks A360015.
Cf. A000070, A000975, A002865, A110618, A237984, A363488.

Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* Gus Wiseman, Jul 10 2023 *)

From Gus Wiseman, Jul 11 2023: (Start)
a(n>2) = A000041(n) - A096373(n-2).
a(n>1) = A000041(n-2) + A002865(n+1).
a(n) = A000041(n+1) - A027336(n).
(End)

Formula corrected by Gus Wiseman, Jul 11 2023

A363954 Numbers whose prime indices have low mean 2.

Original entry on oeis.org

3, 9, 10, 14, 15, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 70, 75, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 189, 196, 198, 204, 208, 210, 220, 225, 234, 243, 250, 252, 260, 264, 270, 272, 280, 294, 297, 300, 304, 308, 312, 315, 330, 350

Offset: 1

Gus Wiseman, Jul 05 2023

nonn

Extending the terminology of A124944, the "low mean" of a multiset is obtained by taking the mean and rounding down.

			The terms together with their prime indices begin:
     3: {2}
     9: {2,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    42: {1,2,4}
    44: {1,1,5}
    45: {2,2,3}
    50: {1,3,3}
    52: {1,1,6}
    63: {2,2,4}
    66: {1,2,5}
    70: {1,3,4}
    75: {2,3,3}
    81: {2,2,2,2}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}

Partitions of this type are counted by A363745.
Positions of 2's in A363943 (high A363944), triangle A363945 (high A363946).
For mean 1 we have A363949.
The high version is A363950, counted by A026905.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
A363948 lists numbers whose prime indices have mean 1, counted by A363947.
Cf. A327473, A327476, A359889, A360013, A360015, A363488, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Floor[Mean[prix[#]]]==2&]

A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

For mean instead of median we have A057716, counted by A000065.
These partitions are counted by A238495.
The complement is A364056, counted by A027336, low version A363488.
A000975 counts subsets with integer median, A051293 for mean.
A124943 counts partitions by low median, high version A124944.
A360005 gives twice the median of prime indices, A360459 for prime factors.
A359893 and A359901 count partitions by median.
Cf. A002865, A316413, A325347, A327473, A327476, A363727.

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

A360005(a(n)) > 1.

A360459(a(n)) > 2.

cp's OEIS Frontend

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

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

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A363941 Low median in the multiset of prime indices of n.

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A363949 Numbers whose prime indices have mean 1 when rounded down.

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A364158 Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.

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A364191 Low co-mode in the multiset of prime indices of n.

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A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

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A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

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A363954 Numbers whose prime indices have low mean 2.

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