cp's OEIS Frontend

A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

%I A360454 #6 Feb 10 2023 17:11:39
%S A360454 1,2,9,54,100,120,125,135,168,180,189,240,252,264,280,297,300,312,336,
%T A360454 351,396,408,440,450,456,459,468,480,513,520,528,540,552,560,588,612,
%U A360454 616,621,624,672,680,684,696,728,744,756,760,783,816,828,837,880,882
%N A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.
%C A360454 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 A360454 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 A360454 The terms together with their prime indices begin:
%e A360454     1: {}
%e A360454     2: {1}
%e A360454     9: {2,2}
%e A360454    54: {1,2,2,2}
%e A360454   100: {1,1,3,3}
%e A360454   120: {1,1,1,2,3}
%e A360454   125: {3,3,3}
%e A360454   135: {2,2,2,3}
%e A360454   168: {1,1,1,2,4}
%e A360454   180: {1,1,2,2,3}
%e A360454   189: {2,2,2,4}
%e A360454   240: {1,1,1,1,2,3}
%e A360454 For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.
%t A360454 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360454 Select[Range[1000],Median[prix[#]]==Median[Length/@Split[prix[#]]]&]
%Y A360454 For mean instead of median we have A359903, counted by A360068.
%Y A360454 For distinct indices instead of indices we have A360453, counted by A360455.
%Y A360454 For distinct indices instead of multiplicities: A360249, counted by A360245.
%Y A360454 These partitions are counted by A360456.
%Y A360454 A088529/A088530 gives mean of prime signature A124010.
%Y A360454 A112798 lists prime indices, length A001222, sum A056239.
%Y A360454 A240219 counts partitions with mean equal to median, ranked by A359889.
%Y A360454 A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
%Y A360454 A326567/A326568 gives mean of prime indices.
%Y A360454 A326619/A326620 gives mean of distinct prime indices.
%Y A360454 A359893 and A359901 count partitions by median.
%Y A360454 A359894 counts partitions with mean different from median, ranks A359890.
%Y A360454 A360005 gives median of prime indices (times two).
%Y A360454 Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.
%K A360454 nonn
%O A360454 1,2
%A A360454 _Gus Wiseman_, Feb 10 2023