%I #6 Jan 23 2023 09:11:00
%S 12,18,20,28,42,44,45,48,50,52,63,66,68,70,72,75,76,78,80,92,98,99,
%T 102,108,112,114,116,117,120,124,130,138,147,148,153,154,162,164,165,
%U 168,170,171,172,174,175,176,180,182,186,188,190,192,195,200,207,208
%N Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.
%C 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.
%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%F Intersection of A026424 and A359890.
%e The terms together with their prime indices begin:
%e 12: {1,1,2}
%e 18: {1,2,2}
%e 20: {1,1,3}
%e 28: {1,1,4}
%e 42: {1,2,4}
%e 44: {1,1,5}
%e 45: {2,2,3}
%e 48: {1,1,1,1,2}
%e 50: {1,3,3}
%e 52: {1,1,6}
%e 63: {2,2,4}
%e 66: {1,2,5}
%e 68: {1,1,7}
%e 70: {1,3,4}
%e 72: {1,1,1,2,2}
%e 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.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],OddQ[PrimeOmega[#]]&&Mean[prix[#]]!=Median[prix[#]]&]
%Y A subset of A026424 = numbers with odd bigomega.
%Y The LHS (mean of prime indices) is A326567/A326568.
%Y This is the odd-length case of A359890, complement A359889.
%Y The complement is A359891.
%Y These partitions are counted by A359896, complement A359895.
%Y The RHS (median of prime indices) is A360005/2.
%Y A058398 counts partitions by mean, see also A008284, A327482.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A316413 lists numbers whose prime indices have integer mean.
%Y A359902 counts odd-length partitions by median.
%Y Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 22 2023