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Low co-mode in the multiset of prime indices of n.
5

%I #16 Oct 18 2023 04:50:18

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

%T 1,3,1,12,1,2,3,13,1,14,5,3,1,15,2,4,1,2,6,16,1,3,4,2,1,17,2,18,1,4,1,

%U 3,1,19,7,2,1,20,2,21,1,2,8,4,1,22,3,2,1

%N Low 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 "low co-mode" in a multiset is the least co-mode.

%F a(n) = A000720(A067695(n)).

%F A067695(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) = 2.

%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,Min[comodes[prix[n]]]],{n,30}]

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

%Y For mode instead of co-mode we have A363486, high A363487, triangle A363952.

%Y For median instead of co-mode we have A363941, high A363942.

%Y Positions of 1's are A364158, counted by A364159.

%Y The high version is A364192 = positions of 1's in A364061.

%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. A124943, A241131, A327473, A327476, A360005, A360015, A363488.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 16 2023