A363487 - OEIS

%I #9 Jul 05 2023 21:37:39

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

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

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

%N High 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 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

%C Extending the terminology of A124944, the "high mode" in a multiset is its greatest mode.

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

%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[If[n==1,0,Last[modes[prix[n]]]],{n,30}]

%Y Positions of first appearances are 1 and A000040.

%Y Positions of 1's are A360015, counted by A241131.

%Y For low instead of high mode we have A363486.

%Y The version for low median is A363941, triangle A124943.

%Y The version for high median is A363942, triangle A124944.

%Y The version for mean instead of mode is A363944, low A363943.

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

%Y A326567/A326568 gives mean of prime indices.

%Y A359178 ranks partitions with a unique co-mode, counted by A362610.

%Y A356862 ranks partitions with a unique mode, counted by A362608.

%Y A362605 ranks partitions with more than one mode, counted by A362607.

%Y A362606 ranks partitions with more than one co-mode, counted by A362609.

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

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

%Y A362616 ranks partitions (max part) = (unique mode), counted by A362612.

%Y Cf. A000961, A215366, A327473, A327476, A359908, A360013, A363723, A363729.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 04 2023