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

%I A359892 #6 Jan 23 2023 09:11:00
%S A359892 12,18,20,28,42,44,45,48,50,52,63,66,68,70,72,75,76,78,80,92,98,99,
%T A359892 102,108,112,114,116,117,120,124,130,138,147,148,153,154,162,164,165,
%U A359892 168,170,171,172,174,175,176,180,182,186,188,190,192,195,200,207,208
%N A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.
%C A359892 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 A359892 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 A359892 Intersection of A026424 and A359890.
%e A359892 The terms together with their prime indices begin:
%e A359892    12: {1,1,2}
%e A359892    18: {1,2,2}
%e A359892    20: {1,1,3}
%e A359892    28: {1,1,4}
%e A359892    42: {1,2,4}
%e A359892    44: {1,1,5}
%e A359892    45: {2,2,3}
%e A359892    48: {1,1,1,1,2}
%e A359892    50: {1,3,3}
%e A359892    52: {1,1,6}
%e A359892    63: {2,2,4}
%e A359892    66: {1,2,5}
%e A359892    68: {1,1,7}
%e A359892    70: {1,3,4}
%e A359892    72: {1,1,1,2,2}
%e A359892 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 A359892 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359892 Select[Range[100],OddQ[PrimeOmega[#]]&&Mean[prix[#]]!=Median[prix[#]]&]
%Y A359892 A subset of A026424 = numbers with odd bigomega.
%Y A359892 The LHS (mean of prime indices) is A326567/A326568.
%Y A359892 This is the odd-length case of A359890, complement A359889.
%Y A359892 The complement is A359891.
%Y A359892 These partitions are counted by A359896, complement A359895.
%Y A359892 The RHS (median of prime indices) is A360005/2.
%Y A359892 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359892 A112798 lists prime indices, length A001222, sum A056239.
%Y A359892 A316413 lists numbers whose prime indices have integer mean.
%Y A359892 A359902 counts odd-length partitions by median.
%Y A359892 Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.
%K A359892 nonn
%O A359892 1,1
%A A359892 _Gus Wiseman_, Jan 22 2023