A367860 Sum of the multiset multiplicity cokernel (in which each multiplicity becomes the greatest element of that multiplicity) of the prime indices of n.

0, 1, 2, 1, 3, 4, 4, 1, 2, 6, 5, 3, 6, 8, 6, 1, 7, 3, 8, 4, 8, 10, 9, 3, 3, 12, 2, 5, 10, 9, 11, 1, 10, 14, 8, 4, 12, 16, 12, 4, 13, 12, 14, 6, 5, 18, 15, 3, 4, 4, 14, 7, 16, 3, 10, 5, 16, 20, 17, 7, 18, 22, 6, 1, 12, 15, 19, 8, 18, 12, 20, 3, 21, 24, 5, 9, 10

Offset: 1

Author

Gus Wiseman, Dec 03 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 the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.

Examples

			The multiset multiplicity cokernel of {1,2,2,3} is {2,3,3}, so a(90) = 8.
		

Crossrefs

Positions of 1's are A000079 without 1.

Depends only on rootless base A052410, see A007916, A052409.

For kernel instead of cokernel we have A367581, row-sums of A367579.

For minimum instead of sum we have A367587, opposite A367583.

The triangle A367858 has these as row sums, ranks A367859.

A007947 gives squarefree kernel.

A112798 lists prime indices, length A001222, sum A056239, reverse A296150.

A124010 gives prime signature, sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A238747 gives prime metasignature, reverse A353742.

A304038 lists distinct prime indices, length A001221, sum A066328.

Cf. A000720, A051904, A055396, A061395, A071625, A072774, A130091, A367580, A175781, A367582, A367584.

Programs

Mathematica

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Total[mmc[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]]],{n,100}]