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Greatest element in row n of A367579 (multiset multiplicity kernel).
12

%I #11 Nov 30 2023 12:25:57

%S 0,1,2,1,3,1,4,1,2,1,5,2,6,1,2,1,7,2,8,3,2,1,9,2,3,1,2,4,10,1,11,1,2,

%T 1,3,1,12,1,2,3,13,1,14,5,3,1,15,2,4,3,2,6,16,2,3,4,2,1,17,2,18,1,4,1,

%U 3,1,19,7,2,1,20,2,21,1,3,8,4,1,22,3,2,1

%N Greatest element in row n of A367579 (multiset multiplicity kernel).

%C 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 We define the multiset multiplicity kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.

%F a(n) = A061395(A367580(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) = A055396(n).

%e For 450 = 2^1 * 3^2 * 5^2, we have MMK({1,2,2,3,3}) = {1,2,2} so a(450) = 2.

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

%t Table[If[n==1,0,Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1,{},FactorInteger[n]]]],{n,1,100}]

%Y Positions of first appearances are A008578.

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

%Y For minimum instead of maximum element we have A055396.

%Y Row maxima of A367579.

%Y Greatest prime index of A367580.

%Y Positions of 1's are A367586 (powers of even squarefree numbers).

%Y The opposite version is A367587.

%Y A007947 gives squarefree kernel.

%Y A072774 lists powers of squarefree numbers.

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

%Y A124010 gives prime signature, sorted A118914.

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

%Y A238747 gives prime metasignature, reverse A353742.

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

%Y A363486 gives least prime index of greatest exponent.

%Y A363487 gives greatest prime index of greatest exponent.

%Y A364191 gives least prime index of least exponent.

%Y A364192 gives greatest prime index of least exponent.

%Y Cf. A000720, A000961, A051904, A061395, A071625, A130091, A289023, A367581, A367584, A367585, A367683.

%K nonn

%O 1,3

%A _Gus Wiseman_, Nov 28 2023