A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882

Offset: 1

Author

Gus Wiseman, Feb 10 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:
    1: {}
    2: {1}
    9: {2,2}
   54: {1,2,2,2}
  100: {1,1,3,3}
  120: {1,1,1,2,3}
  125: {3,3,3}
  135: {2,2,2,3}
  168: {1,1,1,2,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.
		

Crossrefs

For mean instead of median we have A359903, counted by A360068.

For distinct indices instead of indices we have A360453, counted by A360455.

For distinct indices instead of multiplicities: A360249, counted by A360245.

These partitions are counted by A360456.

A088529/A088530 gives mean of prime signature A124010.

A112798 lists prime indices, length A001222, sum A056239.

A240219 counts partitions with mean equal to median, ranked by A359889.

A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

A359893 and A359901 count partitions by median.

A359894 counts partitions with mean different from median, ranks A359890.

A360005 gives median of prime indices (times two).

Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.

Programs

Mathematica

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