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

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

Offset: 1

Author

Gus Wiseman, Jul 16 2023

Keywords

• nonn

Comments

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.

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}.

Extending the terminology of A124943, the "low co-mode" in a multiset is the least co-mode.

Examples

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

Crossrefs

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

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

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

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

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

A112798 lists prime indices, length A001222, sum A056239.

A362611 counts modes in prime indices, triangle A362614.

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

Ranking and counting partitions:

- A356862 = unique mode, counted by A362608

- A359178 = unique co-mode, counted by A362610

- A362605 = multiple modes, counted by A362607

- A362606 = multiple co-modes, counted by A362609

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Min[comodes[prix[n]]]],{n,30}]

Formula

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

A067695(n) = A000040(a(n)).