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%I #6 Feb 10 2023 17:11:39
%S 1,2,9,54,100,120,125,135,168,180,189,240,252,264,280,297,300,312,336,
%T 351,396,408,440,450,456,459,468,480,513,520,528,540,552,560,588,612,
%U 616,621,624,672,680,684,696,728,744,756,760,783,816,828,837,880,882
%N Numbers for which the prime multiplicities (or sorted signature) have the same median as the 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.
%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 1: {}
%e 2: {1}
%e 9: {2,2}
%e 54: {1,2,2,2}
%e 100: {1,1,3,3}
%e 120: {1,1,1,2,3}
%e 125: {3,3,3}
%e 135: {2,2,2,3}
%e 168: {1,1,1,2,4}
%e 180: {1,1,2,2,3}
%e 189: {2,2,2,4}
%e 240: {1,1,1,1,2,3}
%e For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1000],Median[prix[#]]==Median[Length/@Split[prix[#]]]&]
%Y For mean instead of median we have A359903, counted by A360068.
%Y For distinct indices instead of indices we have A360453, counted by A360455.
%Y For distinct indices instead of multiplicities: A360249, counted by A360245.
%Y These partitions are counted by A360456.
%Y A088529/A088530 gives mean of prime signature A124010.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A240219 counts partitions with mean equal to median, ranked by A359889.
%Y A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
%Y A326567/A326568 gives mean of prime indices.
%Y A326619/A326620 gives mean of distinct prime indices.
%Y A359893 and A359901 count partitions by median.
%Y A359894 counts partitions with mean different from median, ranks A359890.
%Y A360005 gives median of prime indices (times two).
%Y Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 10 2023
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