cp's OEIS Frontend

A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

%I A360453 #9 Feb 10 2023 17:11:44
%S A360453 1,2,9,12,18,40,100,112,125,180,250,252,300,352,360,392,396,405,450,
%T A360453 468,504,540,588,600,612,675,684,720,756,792,828,832,882,900,936,1008,
%U A360453 1044,1116,1125,1176,1188,1200,1224,1332,1350,1368,1372,1404,1440,1452,1476
%N A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.
%C A360453 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 A360453 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 A360453 The terms together with their prime indices begin:
%e A360453     1: {}
%e A360453     2: {1}
%e A360453     9: {2,2}
%e A360453    12: {1,1,2}
%e A360453    18: {1,2,2}
%e A360453    40: {1,1,1,3}
%e A360453   100: {1,1,3,3}
%e A360453   112: {1,1,1,1,4}
%e A360453   125: {3,3,3}
%e A360453   180: {1,1,2,2,3}
%e A360453   250: {1,3,3,3}
%e A360453   252: {1,1,2,2,4}
%e A360453   300: {1,1,2,3,3}
%e A360453   352: {1,1,1,1,1,5}
%e A360453   360: {1,1,1,2,2,3}
%e A360453 For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.
%t A360453 Select[Range[100],#==1||Median[Last/@FactorInteger[#]]== Median[PrimePi/@First/@FactorInteger[#]]&]
%Y A360453 Without taking median we have A109298, unordered A109297.
%Y A360453 For mean instead of median we have A324570, counted by A114638.
%Y A360453 For indices instead of multiplicities we have A360249, counted by A360245.
%Y A360453 For indices instead of distinct indices we have A360454, counted by A360456.
%Y A360453 These partitions are counted by A360455.
%Y A360453 A088529/A088530 gives mean of prime signature A124010.
%Y A360453 A112798 lists prime indices, length A001222, sum A056239.
%Y A360453 A240219 counts partitions with mean equal to median, ranks A359889.
%Y A360453 A316413 = numbers whose prime indices have integer mean, distinct A326621.
%Y A360453 A325347 = partitions with integer median, strict A359907, ranks A359908.
%Y A360453 A326567/A326568 gives mean of prime indices.
%Y A360453 A326619/A326620 gives mean of distinct prime indices.
%Y A360453 A359893 and A359901 count partitions by median.
%Y A360453 A360005 gives median of prime indices (times two).
%Y A360453 Cf. A000975, A359890, A359903, A360068, A360244, A360247, A360248.
%K A360453 nonn
%O A360453 1,2
%A A360453 _Gus Wiseman_, Feb 10 2023