A359889 - OEIS

%I #7 Jan 23 2023 09:10:47

%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,

%T 32,33,34,35,36,37,38,39,41,43,46,47,49,51,53,55,57,58,59,61,62,64,65,

%U 67,69,71,73,74,77,79,81,82,83,85,86,87,89,90,91,93,94

%N Numbers that are 1 or whose prime indices have the same mean as median.

%C First differs from A236510 in having 252 (prime indices {1,1,2,2,4}).

%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).

%F Numbers n such that A326567(n)/A326568(n) = A360005(n)/2.

%e The prime indices of 900 are {1,1,2,2,3,3}, with mean 2 and median 2, so 900 is in the sequence.

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

%t Select[Range[100],#==1||Mean[prix[#]]==Median[prix[#]]&]

%Y These partitions are counted by A240219, strict A359897.

%Y The LHS (mean of prime indices) is A326567/A326568.

%Y The complement is A359890, counted by A359894.

%Y The odd-length case is A359891, complement A359892, counted by A359895.

%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 A359893 and A359901 count partitions by median, odd-length A359902.

%Y A359908 lists numbers whose prime indices have integer median.

%Y Cf. A026424, A327473, A359899, A359903, A359905, A359909, A360006, A360007, A360008, A360009.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 22 2023