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%I #6 Jan 23 2023 09:10:52
%S 12,18,20,24,28,40,42,44,45,48,50,52,54,56,60,63,66,68,70,72,75,76,78,
%T 80,84,88,92,96,98,99,102,104,108,112,114,116,117,120,124,126,130,132,
%U 135,136,138,140,144,147,148,150,152,153,154,156,160,162,164,165
%N Numbers 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).
%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 24: {1,1,1,2}
%e 28: {1,1,4}
%e 40: {1,1,1,3}
%e 42: {1,2,4}
%e 44: {1,1,5}
%e 45: {2,2,3}
%e 48: {1,1,1,1,2}
%e For example, the prime indices of 360 are {1,1,1,2,2,3}, with mean 5/3 and median 3/2, so 360 is in the sequence.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1000],Mean[prix[#]]!=Median[prix[#]]&]
%Y The LHS (mean of prime indices) is A326567/A326568.
%Y The complement is A359889, counted by A240219.
%Y The odd-length case is A359891, complement A359892.
%Y These partitions are counted by A359894.
%Y The strict case is counted by A359898, odd-length A359900.
%Y The RHS (median of prime indices) is A360005/2.
%Y A058398 counts partitions by mean, see also A008284, A327482.
%Y A088529/A088530 gives mean of prime signature A124010.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A316413 lists numbers whose prime indices have integer mean.
%Y A359908 lists numbers whose prime indices have integer median.
%Y Cf. A327473, A327476, A348551, A359903, A359911, A359912, A360006-A360009.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 22 2023