cp's OEIS Frontend

A364191 Low co-mode in the multiset of prime indices of n.

%I A364191 #16 Oct 18 2023 04:50:18
%S A364191 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 A364191 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 A364191 3,1,19,7,2,1,20,2,21,1,2,8,4,1,22,3,2,1
%N A364191 Low co-mode in the multiset of prime indices of n.
%C A364191 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 A364191 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 A364191 Extending the terminology of A124943, the "low co-mode" in a multiset is the least co-mode.
%F A364191 a(n) = A000720(A067695(n)).
%F A364191 A067695(n) = A000040(a(n)).
%e A364191 The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 2.
%t A364191 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A364191 comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
%t A364191 Table[If[n==1,0,Min[comodes[prix[n]]]],{n,30}]
%Y A364191 For prime factors instead of indices we have A067695, high A359612.
%Y A364191 For mode instead of co-mode we have A363486, high A363487, triangle A363952.
%Y A364191 For median instead of co-mode we have A363941, high A363942.
%Y A364191 Positions of 1's are A364158, counted by A364159.
%Y A364191 The high version is A364192 = positions of 1's in A364061.
%Y A364191 A112798 lists prime indices, length A001222, sum A056239.
%Y A364191 A362611 counts modes in prime indices, triangle A362614.
%Y A364191 A362613 counts co-modes in prime indices, triangle A362615.
%Y A364191 Ranking and counting partitions:
%Y A364191 - A356862 = unique mode, counted by A362608
%Y A364191 - A359178 = unique co-mode, counted by A362610
%Y A364191 - A362605 = multiple modes, counted by A362607
%Y A364191 - A362606 = multiple co-modes, counted by A362609
%Y A364191 Cf. A124943, A241131, A327473, A327476, A360005, A360015, A363488.
%K A364191 nonn
%O A364191 1,3
%A A364191 _Gus Wiseman_, Jul 16 2023