%I #7 Feb 10 2023 14:29:29

%S 12,20,24,28,40,44,45,48,52,56,60,63,68,72,76,80,84,88,92,96,99,104,

%T 112,116,117,120,124,126,132,135,136,140,144,148,152,153,156,160,164,

%U 168,171,172,175,176,180,184,188,189,192,198,200,204,207,208,212,220

%N Numbers for which the prime indices have lesser mean than the distinct prime indices.

%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.

%e The terms together with their prime indices begin:

%e 12: {1,1,2}

%e 20: {1,1,3}

%e 24: {1,1,1,2}

%e 28: {1,1,4}

%e 40: {1,1,1,3}

%e 44: {1,1,5}

%e 45: {2,2,3}

%e 48: {1,1,1,1,2}

%e 52: {1,1,6}

%e 56: {1,1,1,4}

%e 60: {1,1,2,3}

%e 63: {2,2,4}

%e 68: {1,1,7}

%e 72: {1,1,1,2,2}

%e For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is not in the sequence.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Mean[prix[#]]<Mean[Union[prix[#]]]&]

%Y These partitions are counted by A360251.

%Y For unequal instead of less we have A360246, counted by A360242.

%Y For equal instead of less we have A360247, counted by A360243.

%Y For greater instead of less we have A360252, counted by A360250.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A316413 lists numbers whose indices have integer mean, distinct A326621.

%Y A326567/A326568 gives mean of prime indices.

%Y A326619/A326620 gives mean of distinct prime indices.

%Y Cf. A000975, A051293, A058398, A067340, A067538, A324570, A327482, A359903, A360005, A360241, A360248.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 09 2023