A359612 -id:A359612

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A364191

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..82.

FORMULA

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

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

EXAMPLE

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

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}]

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 16 2023

STATUS

approved

A364192

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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 "high co-mode" in a multiset is the greatest co-mode.

LINKS

Table of n, a(n) for n=1..81.

FORMULA

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

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

EXAMPLE

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