A360248 - OEIS

%I #7 Feb 09 2023 20:49:19

%S 12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,92,

%T 96,98,99,104,108,112,116,117,120,124,132,135,136,140,144,147,148,150,

%U 152,153,156,160,162,164,168,171,172,175,176,184,188,189,192,200

%N Numbers for which the prime indices do not have the same median as the distinct prime indices.

%C First differs from A242416 in lacking 180, with prime indices {1,1,2,2,3}.

%C First differs from A360246 in lacking 126 and having 1950.

%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   12: {1,1,2}

%e   18: {1,2,2}

%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   50: {1,3,3}

%e   52: {1,1,6}

%e   54: {1,2,2,2}

%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 The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is not in the sequence.

%e The prime indices of 1950 are {1,2,3,3,6} with median 3 and distinct prime indices {1,2,3,6} with median 5/2, so 1950 is in the sequence.

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

%t Select[Range[100],Median[prix[#]]!=Median[Union[prix[#]]]&]

%Y These partitions are counted by A360244.

%Y The complement is A360249, counted by A360245.

%Y For multiplicities instead of parts: complement of A360453.

%Y For multiplicities instead of distinct parts: complement of A360454.

%Y For mean instead of median we have A360246, counted by A360242.

%Y The complement for mean instead of median is A360247, counted by A360243.

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

%Y A326567/A326568 gives mean of prime indices.

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

%Y A325347 = partitions with integer median, strict A359907, ranked by A359908.

%Y A359893 and A359901 count partitions by median.

%Y A360005 gives median of prime indices (times two).

%Y Cf. A000975, A078174, A316413, A324570, A359890, A360455, A360456.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 07 2023