%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