A359908 -id:A359908 - cp's OEIS frontend

A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154

Offset: 1

Gus Wiseman, May 05 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The prime indices of 180 are {1,1,2,2,3}, with modes {1,2}, so 180 is in the sequence, and the sequence differs from A182853.
The terms together with their prime indices begin:
     6: {1,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    21: {2,4}
    22: {1,5}
    26: {1,6}
    30: {1,2,3}
    33: {2,5}
    34: {1,7}
    35: {3,4}
    36: {1,1,2,2}
    38: {1,8}
    39: {2,6}
    42: {1,2,4}
    46: {1,9}
    51: {2,7}
    55: {3,5}

Amiram Eldar, Table of n, a(n) for n = 1..10000

The first term with bigomega n appears to be A166023(n).
The complement is A356862, counted by A362608.
For co-mode complement we have A359178, counted by A362610.
For co-mode we have A362606, counted by A362609.
Partitions of this type are counted by A362607.
These are the positions of terms > 1 in A362611.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes, ranks A362611.
A362615 counts partitions by number of co-modes, ranks A362613.
Cf. A002865, A215366, A327473, A327476, A353864, A359908, A362612.

q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):
select(q, [$1..250])[];  # Alois P. Heinz, May 10 2023

Select[Range[100],Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]

is(n) = {my(e = factor(n)[, 2]); if(#e < 2, 0, e = vecsort(e); e[#e-1] == e[#e]);} \\ Amiram Eldar, Jan 20 2024

from sympy import factorint
def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1
print([k for k in range(155) if ok(k)]) # Michael S. Branicky, May 06 2023

A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A130092 in lacking 180.
First differs from A351295 in lacking 180 and having 216.
First differs from A362605 in having 60.

			The prime factorization of 1800 is {2,2,2,3,3,5,5}, and the parts of least multiplicity are {3,5}, so 1800 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}

The complement is A359178, counted by A362610.
For mode we have A362605, counted by A362607.
Partitions of this type are counted by A362609.
These are the positions of terms > 1 in A362613.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes.
A362615 counts partitions by number of co-modes.
Cf. A215366, A327473, A327476, A353864, A356862, A359908, A362611.

Select[Range[100],Count[Last/@FactorInteger[#],Min@@Last/@FactorInteger[#]]>1&]

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.

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)).

A238479 Number of partitions of n whose median is not a part.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 8, 10, 13, 18, 23, 30, 40, 50, 64, 83, 104, 131, 166, 206, 256, 320, 394, 485, 598, 730, 891, 1088, 1318, 1596, 1932, 2326, 2797, 3360, 4020, 4804, 5735, 6824, 8108, 9624, 11392, 13468, 15904, 18737, 22048, 25914, 30400, 35619, 41686

Offset: 1

Clark Kimberling, Feb 27 2014

Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.

Also the number of even-length integer partitions of n with distinct middle parts. For example, the partition (4,3,2,1) has middle parts {2,3} so is counted under a(10), but (3,2,2,1) has middle parts {2,2} so is not counted under a(8). - Gus Wiseman, May 13 2023

			a(6) counts these partitions:  51, 42, 2211 which all have an even number of parts, and their medians 3, 3 and 1.5 are not present. Note that the partitions 33 and 3111, although having an even number of parts, are not included in the count of a(6), but instead in that of A238478(6), as their medians, 3 for the former and 1 for the latter, are present in those partitions.

A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

The complement is A238478, ranks A362618.
For mean instead of median we have A327472, complement A237984.
These partitions have ranks A362617.
A000041 counts integer partitions, even-length A027187.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
A359908 ranks partitions with integer median, complement A359912.
Cf. A002865, A008284, A053263, A238480, A238481, A240219, A359907, A360686, A360687, A360952.

Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]

my(q='q+O('q^50)); concat([0,0], Vec(sum(n=1,17,q^(3*n)/prod(k=1,2*n,1-q^k)))) \\ David Radcliffe, Jun 25 2025

from sympy.utilities.iterables import partitions
def A238479(n): return sum(1 for p in partitions(n) if (m:=max(p,default=0))&1^1 and m>>1 in p) # Chai Wah Wu, Sep 21 2023

a(n) = A000041(n) - A238478(n).
For all n, A027187(n) >= a(n). [Because when a partition of n has an odd number of parts, then it is not counted by this sequence (cf. A238478) and also some of the partitions with an even number of parts might be excluded here. Cf. Examples.] - Antti Karttunen, Feb 27 2014
From Jeremy Lovejoy, Sep 29 2022: (Start)
G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).
a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)
a(n) = A087897(2*n) = A035294(n) - A078408(n-1). - Mathew Englander, May 20 2023

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[#]]]&]

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 *)

A363487 High mode in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 04 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}.
Extending the terminology of A124944, the "high mode" in a multiset is its greatest mode.

Positions of first appearances are 1 and A000040.
Positions of 1's are A360015, counted by A241131.
For low instead of high mode we have A363486.
The version for low median is A363941, triangle A124943.
The version for high median is A363942, triangle A124944.
The version for mean instead of mode is A363944, low A363943.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A359908, A360013, A363723, A363729.

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]&];
Table[If[n==1,0,Last[modes[prix[n]]]],{n,30}]

cp's OEIS Frontend

A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.

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A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

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

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

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

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

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A238479 Number of partitions of n whose median is not a part.

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

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