A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 3, 6, 2, 4, 1, 7, 3, 8, 4, 4, 2, 9, 3, 3, 2, 2, 5, 10, 3, 11, 1, 4, 2, 6, 2, 12, 2, 4, 4, 13, 3, 14, 6, 5, 2, 15, 3, 4, 4, 4, 7, 16, 3, 6, 5, 4, 2, 17, 5, 18, 2, 6, 1, 6, 3, 19, 8, 4, 3, 20, 3, 21, 2, 5, 9, 8, 3, 22, 4, 2, 2

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}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

Examples

			The multiset multiplicity kernel of {1,2,2,3} is {1,1,2}, so a(90) = 4.
		

Crossrefs

Positions of 1's are A000079 without 1.

Positions of first appearances are A008578.

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

The triangle A367579 has these as row sums, ranks A367580.

The triangle for this rank statistic is A367582.

For maximum instead of sum we have A367583, opposite A367587.

A007947 gives squarefree kernel.

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.

Cf. A000720, A005117, A051904, A055396, A061395, A071625, A072774, A130091, A175781, A367584, A367585.

Programs

Mathematica

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

Formula

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

a(n) = A056239(A367580(n)).

If n is squarefree, a(n) = A055396(n)*A001222(n).