A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

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

Offset: 1

Author

Gus Wiseman, Dec 03 2023

Keywords

• nonn

Comments

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.

Crossrefs

Indices of first appearances are A008578.

Depends only on rootless base A052410, see A007916.

For kernel instead of cokernel we have A055396.

For maximum instead of minimum element we have A061395.

The opposite version is A367583.

Row-minima of A367858.

A007947 gives squarefree kernel.

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

A124010 lists prime multiplicities (prime signature), sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A238747 gives prime metasignature, sorted A353742.

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

A367579 lists MMK, rank A367580, sum A367581, max A367583, min A055396.

Cf. A000720, A027746, A051904, A071625, A072774, A130091, A367582, A367584, A367585, A367586.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Min@@mmc[prix[n]]],{n,100}]

Formula

a(n) = A055396(A367859(n)).

a(n^k) = a(n) for all positive integers n and k.

If n is a power of a squarefree number, a(n) = A061395(n).