A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

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

Offset: 1

Author

Gus Wiseman, Nov 28 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 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}.

Examples

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

Crossrefs

Positions of first appearances are A008578.

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

For minimum instead of maximum element we have A055396.

Row maxima of A367579.

Greatest prime index of A367580.

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

The opposite version is A367587.

A007947 gives squarefree kernel.

A072774 lists powers of squarefree numbers.

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.

A363486 gives least prime index of greatest exponent.

A363487 gives greatest prime index of greatest exponent.

A364191 gives least prime index of least exponent.

A364192 gives greatest prime index of least exponent.

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

Programs

Mathematica

mmk[q_]:=With[{mts=Length/@Split[q]},Sort[Table[Min@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1,{},FactorInteger[n]]]],{n,1,100}]

Formula

a(n) = A061395(A367580(n)).

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

If n is a power of a squarefree number, a(n) = A055396(n).