A359891 Members of A026424 (numbers with an odd number of prime factors) whose prime indices have the same mean as median.

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 30, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 110, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Author

Gus Wiseman, Jan 22 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:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   8: {1,1,1}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  23: {9}
  27: {2,2,2}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is not in the sequence.
		

Crossrefs

A subset of A026424 = numbers with odd bigomega.

The LHS (mean of prime indices) is A326567/A326568.

This is the odd-length case of A359889, complement A359890.

The complement is A359892.

These partitions are counted by A359895, any-length A240219.

The RHS (median of prime indices) is A360005/2.

A058398 counts partitions by mean, see also A008284, A327482.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose prime indices have integer mean.

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

A359908 lists numbers whose prime indices have integer median.

Cf. A327473, A359894, A359899, A359910, A360007, A360009.

Programs

Mathematica

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

Formula

Intersection of A026424 and A359889.