A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

1, 2, 9, 12, 18, 40, 100, 112, 125, 180, 250, 252, 300, 352, 360, 392, 396, 405, 450, 468, 504, 540, 588, 600, 612, 675, 684, 720, 756, 792, 828, 832, 882, 900, 936, 1008, 1044, 1116, 1125, 1176, 1188, 1200, 1224, 1332, 1350, 1368, 1372, 1404, 1440, 1452, 1476

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}
   12: {1,1,2}
   18: {1,2,2}
   40: {1,1,1,3}
  100: {1,1,3,3}
  112: {1,1,1,1,4}
  125: {3,3,3}
  180: {1,1,2,2,3}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  300: {1,1,2,3,3}
  352: {1,1,1,1,1,5}
  360: {1,1,1,2,2,3}
For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.
		

Crossrefs

Without taking median we have A109298, unordered A109297.

For mean instead of median we have A324570, counted by A114638.

For indices instead of multiplicities we have A360249, counted by A360245.

For indices instead of distinct indices we have A360454, counted by A360456.

These partitions are counted by A360455.

A088529/A088530 gives mean of prime signature A124010.

A112798 lists prime indices, length A001222, sum A056239.

A240219 counts partitions with mean equal to median, ranks A359889.

A316413 = numbers whose prime indices have integer mean, distinct A326621.

A325347 = partitions with 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.

A360005 gives median of prime indices (times two).

Cf. A000975, A359890, A359903, A360068, A360244, A360247, A360248.

Programs

Mathematica

Select[Range[100],#==1||Median[Last/@FactorInteger[#]]== Median[PrimePi/@First/@FactorInteger[#]]&]