login
Least element in row n of A367858 (multiset multiplicity cokernel).
8

%I #6 Dec 03 2023 23:30:56

%S 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,

%T 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,

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

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

%C 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.

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

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

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

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&],{i,mts}]]];

%t Table[If[n==1,0,Min@@mmc[prix[n]]],{n,100}]

%Y Indices of first appearances are A008578.

%Y Depends only on rootless base A052410, see A007916.

%Y For kernel instead of cokernel we have A055396.

%Y For maximum instead of minimum element we have A061395.

%Y The opposite version is A367583.

%Y Row-minima of A367858.

%Y A007947 gives squarefree kernel.

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

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

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A238747 gives prime metasignature, sorted A353742.

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

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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Dec 03 2023