A364192 - OEIS

%I #14 Oct 18 2023 04:50:42

%S 0,1,2,1,3,2,4,1,2,3,5,2,6,4,3,1,7,1,8,3,4,5,9,2,3,6,2,4,10,3,11,1,5,

%T 7,4,2,12,8,6,3,13,4,14,5,3,9,15,2,4,1,7,6,16,1,5,4,8,10,17,3,18,11,4,

%U 1,6,5,19,7,9,4,20,2,21,12,2,8,5,6,22,3,2

%N High (i.e., greatest) co-mode in the multiset of prime indices of n.

%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 We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.

%C Extending the terminology of A124943, the "high co-mode" in a multiset is the greatest co-mode.

%F a(n) = A000720(A359612(n)).

%F A359612(n) = A000040(a(n)).

%e The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 4.

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

%t comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];

%t Table[If[n==1,0,Max[comodes[prix[n]]]],{n,30}]

%Y For prime factors instead of indices we have A359612, low A067695.

%Y For mode instead of co-mode we have A363487 (triangle A363953), low A363486 (triangle A363952).

%Y The version for median instead of co-mode is A363942, low A363941.

%Y Positions of 1's are A364061, counted by A364062.

%Y The low version is A364191, 1's at A364158 (counted by A364159).

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

%Y A362611 counts modes in prime indices, triangle A362614.

%Y A362613 counts co-modes in prime indices, triangle A362615.

%Y Ranking and counting partitions:

%Y - A356862 = unique mode, counted by A362608

%Y - A359178 = unique co-mode, counted by A362610

%Y - A362605 = multiple modes, counted by A362607

%Y - A362606 = multiple co-modes, counted by A362609

%Y Cf. A241131, A327473, A327476, A360005, A360015, A362612, A363740.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 16 2023