A359900 -id:A359900 - cp's OEIS frontend

A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

12, 18, 20, 28, 42, 44, 45, 48, 50, 52, 63, 66, 68, 70, 72, 75, 76, 78, 80, 92, 98, 99, 102, 108, 112, 114, 116, 117, 120, 124, 130, 138, 147, 148, 153, 154, 162, 164, 165, 168, 170, 171, 172, 174, 175, 176, 180, 182, 186, 188, 190, 192, 195, 200, 207, 208

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}
   28: {1,1,4}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is in the sequence.

A subset of A026424 = numbers with odd bigomega.
The LHS (mean of prime indices) is A326567/A326568.
This is the odd-length case of A359890, complement A359889.
The complement is A359891.
These partitions are counted by A359896, complement A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359902 counts odd-length partitions by median.
Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.

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

Intersection of A026424 and A359890.

A363721 Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 2, 5, 7, 1, 2, 8, 2, 9, 16, 11, 2, 2, 15, 16, 37, 33, 2, 44, 2, 1, 79, 33, 103, 127, 2, 47, 166, 39, 2, 214, 2, 384, 738, 90, 2, 2, 277, 185, 631, 1077, 2, 1065, 1560, 477, 1156, 223, 2, 2863

Offset: 1

Gus Wiseman, Jun 21 2023

nonn

The median of an odd-length partition is the middle part.

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 a(n) partitions for n = {1, 3, 9, 14, 15, 18, 20, 22} (A..M = 10..22):
  1  3    9          E        F                I          K      M
     111  333        2222222  555              666        44444  22222222222
          111111111  3222221  33333            222222222  54443  32222222221
                     3322211  43332            322222221  64442  33222222211
                     4222211  53331            332222211  65441  33322222111
                              63321            422222211  74432  42222222211
                              111111111111111  432222111  74441  43222222111
                                               522222111  84431  44222221111
                                                          94421  52222222111
                                                                 53222221111
                                                                 62222221111

All odd-length partitions are counted by A027193.
For just (mean) = (median) we have A359895, also A240219, A359899, A359910.
For just (mean) != (median) we have A359896, also A359894, A359900.
Allowing any length gives A363719, ranks A363727, non-constant A363728.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
A363726 counts odd-length partitions with a unique mode.
Cf. A237984, A325347, A326567/A326568, A327472, A363720, A363740, A363741.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

cp's OEIS Frontend

A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

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