A363223 Numbers with bigomega equal to median prime index.

2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970

Offset: 1

Author

Gus Wiseman, May 29 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

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.

Examples

			The terms together with their prime indices begin:
    2: {1}
    9: {2,2}
   10: {1,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  130: {1,3,6}
  165: {2,3,5}
  170: {1,3,7}
  175: {3,3,4}
		

Crossrefs

For maximum instead of median we have A106529, counted by A047993.

For minimum instead of median we have A324522, counted by A006141.

Partitions of this type are counted by A361800.

For twice median we have A362050, counted by A362049.

For maximum instead of length we have A362621, counted by A053263.

A000975 counts subsets with integer median.

A027746 lists prime factors, A112798 indices, length A001222, sum A056239.

A325347 counts partitions with integer median, complement A307683.

A359893 and A359901 count partitions by median.

A359908 lists numbers whose prime indices have integer median.

A360005 gives twice median of prime indices.

Cf. A000040, A013580, A079309, A240219, A327473, A327476, A361860, A362619, A362622, A362980.

Programs

Mathematica

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

Formula

2*A001222(a(n)) = A360005(a(n)).