A363487 High mode in the multiset of prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Jul 04 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.

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}.

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

Crossrefs

Positions of first appearances are 1 and A000040.

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

For low instead of high mode we have A363486.

The version for low median is A363941, triangle A124943.

The version for high median is A363942, triangle A124944.

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

A112798 lists prime indices, length A001222, sum A056239.

A326567/A326568 gives mean of prime indices.

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

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

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

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

A362611 counts modes in prime indices, triangle A362614.

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

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

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

Programs

Mathematica

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