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Numbers whose prime indices do not have the same mean as median.
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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