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%I #5 Jun 24 2023 13:03:02
%S 42,60,66,70,78,84,102,114,130,132,138,140,150,154,156,165,170,174,
%T 180,182,186,190,195,204,220,222,228,230,231,246,255,258,260,266,276,
%U 282,285,286,290,294,308,310,315,318,322,330,340,345,348,354,357,360,364
%N Numbers whose prime indices have different mean, median, and mode.
%C If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
%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 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
%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 All three of A326567(a(n))/A326568(a(n)), A360005(a(n))/2, and A363486(a(n)) = A363487(a(n)) are different.
%e The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, median 2, modes {1,2}, so 180 is in the sequence.
%e The prime indices of 108 are {1,1,2,2,2}, with mean 8/5, median 2, modes {2}, so 108 is not in the sequence.
%e The terms together with their prime indices begin:
%e 42: {1,2,4}
%e 60: {1,1,2,3}
%e 66: {1,2,5}
%e 70: {1,3,4}
%e 78: {1,2,6}
%e 84: {1,1,2,4}
%e 102: {1,2,7}
%e 114: {1,2,8}
%e 130: {1,3,6}
%e 132: {1,1,2,5}
%e 138: {1,2,9}
%e 140: {1,1,3,4}
%e 150: {1,2,3,3}
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t Select[Range[100],{Mean[prix[#]]}!={Median[prix[#]]}!=modes[prix[#]]&]
%Y These partitions are counted by A363720
%Y For equal instead of unequal we have A363727, counted by A363719.
%Y The version for factorizations is A363742, equal A363741.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A326567/A326568 gives mean of prime indices.
%Y A356862 ranks partitions with a unique mode, counted by A362608.
%Y A359178 ranks partitions with multiple modes, counted by A362610.
%Y A360005 gives twice the median of prime indices.
%Y A362611 counts modes in prime indices, triangle A362614.
%Y A362613 counts co-modes in prime indices, triangle A362615.
%Y A363486 gives least mode in prime indices, A363487 greatest.
%Y Just two statistics:
%Y - (mean) = (median): A359889, counted by A240219.
%Y - (mean) != (median): A359890, counted by A359894.
%Y - (mean) = (mode): counted by A363723, see A363724, A363731.
%Y - (median) = (mode): counted by A363740.
%Y Cf. A000961, A327473, A327476, A359908, A363722, A363725, A363729.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 24 2023
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