A360009 - OEIS

%I #6 Jan 24 2023 12:35:30

%S 2,3,4,5,7,8,9,10,11,13,16,17,19,21,22,23,25,27,28,29,30,31,32,34,37,

%T 39,41,43,46,47,49,53,55,57,59,61,62,64,67,68,71,73,78,79,81,82,83,85,

%U 87,88,89,90,91,94,97,98,99,100,101,103,105,107,109,110,111

%N Numbers whose prime indices have integer mean and integer median.

%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 Intersection of A316413 and A359908.

%e The terms together with their prime indices begin:

%e     2: {1}

%e     3: {2}

%e     4: {1,1}

%e     5: {3}

%e     7: {4}

%e     8: {1,1,1}

%e     9: {2,2}

%e    10: {1,3}

%e    11: {5}

%e    13: {6}

%e    16: {1,1,1,1}

%e    17: {7}

%e    19: {8}

%e    21: {2,4}

%e    22: {1,5}

%e    23: {9}

%e    25: {3,3}

%e    27: {2,2,2}

%e    28: {1,1,4}

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

%t Select[Range[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Median[prix[#]]]&]

%Y For just integer mean we have A316413 (counted by A067538).

%Y The mean of prime indices is given by A326567/A326568.

%Y The complement is A348551 \/ A359912 (counted by A349156 and A307683).

%Y These partitions are counted by A359906.

%Y For just integer median we have A359908 (counted by A325347).

%Y The median of prime indices is given by A360005/2.

%Y A058398 counts partitions by mean, see also A008284, A327482.

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

%Y A326622 counts factorizations with integer mean, strict A328966.

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

%Y Cf. A026424, A327473, A359889, A359890, A359903, A359905, A360006.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 24 2023