A359908 Numbers whose prime indices have integer median.

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 82, 83

Offset: 1

Author

Gus Wiseman, Jan 23 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 prime indices of 180 are {1,1,2,2,3}, with median 2, so 180 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with median 3/2, so 360 is not in the sequence.
		

Crossrefs

The odd-length case is A027193.

For mean instead of median we have A316413.

These partitions are counted by A325347, strict A359907.

The complement is A359912, counted by A307683.

The median of prime indices is given by A360005/2.

The case of integer mean also is A360009.

A112798 lists prime indices, length A001222, sum A056239.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A026424, A359889, A359890, A359903, A359905, A360006.

Programs

Mathematica

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