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

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

Author

Gus Wiseman, Feb 07 2023

Keywords

• nonn

Comments

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).

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}
  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.
		

Crossrefs

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.

Programs

Mathematica

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