cp's OEIS Frontend

A360009 Numbers whose prime indices have integer mean and integer median.

%I A360009 #6 Jan 24 2023 12:35:30
%S A360009 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 A360009 39,41,43,46,47,49,53,55,57,59,61,62,64,67,68,71,73,78,79,81,82,83,85,
%U A360009 87,88,89,90,91,94,97,98,99,100,101,103,105,107,109,110,111
%N A360009 Numbers whose prime indices have integer mean and integer median.
%C A360009 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 A360009 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 A360009 Intersection of A316413 and A359908.
%e A360009 The terms together with their prime indices begin:
%e A360009     2: {1}
%e A360009     3: {2}
%e A360009     4: {1,1}
%e A360009     5: {3}
%e A360009     7: {4}
%e A360009     8: {1,1,1}
%e A360009     9: {2,2}
%e A360009    10: {1,3}
%e A360009    11: {5}
%e A360009    13: {6}
%e A360009    16: {1,1,1,1}
%e A360009    17: {7}
%e A360009    19: {8}
%e A360009    21: {2,4}
%e A360009    22: {1,5}
%e A360009    23: {9}
%e A360009    25: {3,3}
%e A360009    27: {2,2,2}
%e A360009    28: {1,1,4}
%t A360009 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360009 Select[Range[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Median[prix[#]]]&]
%Y A360009 For just integer mean we have A316413 (counted by A067538).
%Y A360009 The mean of prime indices is given by A326567/A326568.
%Y A360009 The complement is A348551 \/ A359912 (counted by A349156 and A307683).
%Y A360009 These partitions are counted by A359906.
%Y A360009 For just integer median we have A359908 (counted by A325347).
%Y A360009 The median of prime indices is given by A360005/2.
%Y A360009 A058398 counts partitions by mean, see also A008284, A327482.
%Y A360009 A112798 lists prime indices, length A001222, sum A056239.
%Y A360009 A326622 counts factorizations with integer mean, strict A328966.
%Y A360009 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360009 Cf. A026424, A327473, A359889, A359890, A359903, A359905, A360006.
%K A360009 nonn
%O A360009 1,1
%A A360009 _Gus Wiseman_, Jan 24 2023