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

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

Author

Gus Wiseman, Jan 22 2023

Keywords

• nonn

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 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.
		

Crossrefs

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.

Programs

Mathematica

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