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A359908 Numbers whose prime indices have integer median.

%I A359908 #5 Jan 23 2023 12:41:33
%S A359908 2,3,4,5,7,8,9,10,11,12,13,16,17,18,19,20,21,22,23,24,25,27,28,29,30,
%T A359908 31,32,34,37,39,40,41,42,43,44,45,46,47,48,49,50,52,53,54,55,56,57,59,
%U A359908 61,62,63,64,66,67,68,70,71,72,73,75,76,78,79,80,81,82,83
%N A359908 Numbers whose prime indices have integer median.
%C A359908 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 A359908 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 A359908 The prime indices of 180 are {1,1,2,2,3}, with median 2, so 180 is in the sequence.
%e A359908 The prime indices of 360 are {1,1,1,2,2,3}, with median 3/2, so 360 is not in the sequence.
%t A359908 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359908 Select[Range[100],IntegerQ[Median[prix[#]]]&]
%Y A359908 The odd-length case is A027193.
%Y A359908 For mean instead of median we have A316413.
%Y A359908 These partitions are counted by A325347, strict A359907.
%Y A359908 The complement is A359912, counted by A307683.
%Y A359908 The median of prime indices is given by A360005/2.
%Y A359908 The case of integer mean also is A360009.
%Y A359908 A112798 lists prime indices, length A001222, sum A056239.
%Y A359908 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A359908 Cf. A026424, A359889, A359890, A359903, A359905, A360006.
%K A359908 nonn
%O A359908 1,1
%A A359908 _Gus Wiseman_, Jan 23 2023