cp's OEIS Frontend

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

%I A367860 #5 Dec 04 2023 06:39:11
%S A367860 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,
%T A367860 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,
%U A367860 17,7,18,22,6,1,12,15,19,8,18,12,20,3,21,24,5,9,10
%N A367860 Sum of the multiset multiplicity cokernel (in which each multiplicity becomes the greatest element of that multiplicity) of the prime indices of n.
%C A367860 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.
%C A367860 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.
%e A367860 The multiset multiplicity cokernel of {1,2,2,3} is {2,3,3}, so a(90) = 8.
%t A367860 mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367860 Table[Total[mmc[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]]],{n,100}]
%Y A367860 Positions of 1's are A000079 without 1.
%Y A367860 Depends only on rootless base A052410, see A007916, A052409.
%Y A367860 For kernel instead of cokernel we have A367581, row-sums of A367579.
%Y A367860 For minimum instead of sum we have A367587, opposite A367583.
%Y A367860 The triangle A367858 has these as row sums, ranks A367859.
%Y A367860 A007947 gives squarefree kernel.
%Y A367860 A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A367860 A124010 gives prime signature, sorted A118914.
%Y A367860 A181819 gives prime shadow, with an inverse A181821.
%Y A367860 A238747 gives prime metasignature, reverse A353742.
%Y A367860 A304038 lists distinct prime indices, length A001221, sum A066328.
%Y A367860 Cf. A000720, A051904, A055396, A061395, A071625, A072774, A130091, A367580, A175781, A367582, A367584.
%K A367860 nonn
%O A367860 1,3
%A A367860 _Gus Wiseman_, Dec 03 2023