A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

12, 18, 20, 28, 42, 44, 45, 48, 50, 52, 63, 66, 68, 70, 72, 75, 76, 78, 80, 92, 98, 99, 102, 108, 112, 114, 116, 117, 120, 124, 130, 138, 147, 148, 153, 154, 162, 164, 165, 168, 170, 171, 172, 174, 175, 176, 180, 182, 186, 188, 190, 192, 195, 200, 207, 208

Offset

1,1

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).

Links

Table of n, a(n) for n=1..56.

Formula

Intersection of A026424 and A359890.

Example

The terms together with their prime indices begin:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   28: {1,1,4}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is in the sequence.

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[#]]&]

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 A359890, complement A359889.

The complement is A359891.

These partitions are counted by A359896, complement A359895.

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.

A359902 counts odd-length partitions by median.

Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.

Sequence in context: A030515 A162947 A351201 * A070011 A084679 A072588

Adjacent sequences: A359889 A359890 A359891 * A359893 A359894 A359895

Keyword

nonn

Author

Gus Wiseman, Jan 22 2023

Status

approved