A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 6, 1, 1, 2, 2, 1, 7, 1, 2, 8, 1, 3, 2, 2, 1, 1, 9, 1, 2, 3, 1, 1, 2, 1, 4, 10, 1, 1, 1, 11, 1, 2, 2, 1, 1, 3, 3, 1, 1, 12, 1, 1, 2, 2, 1, 3, 13, 1, 1, 1, 14, 1, 5, 2, 3, 1, 1, 15, 1, 2, 4, 1, 3, 2, 2, 1, 6, 16, 1, 2

Offset: 1

Author

Gus Wiseman, Nov 25 2023

Keywords

• nonn
• tabf

Comments

Row n = 1 is empty.

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}.

Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.

Examples

			The first 45 rows:
     1: {}      16: {1}       31: {11}
     2: {1}     17: {7}       32: {1}
     3: {2}     18: {1,2}     33: {2,2}
     4: {1}     19: {8}       34: {1,1}
     5: {3}     20: {1,3}     35: {3,3}
     6: {1,1}   21: {2,2}     36: {1,1}
     7: {4}     22: {1,1}     37: {12}
     8: {1}     23: {9}       38: {1,1}
     9: {2}     24: {1,2}     39: {2,2}
    10: {1,1}   25: {3}       40: {1,3}
    11: {5}     26: {1,1}     41: {13}
    12: {1,2}   27: {2}       42: {1,1,1}
    13: {6}     28: {1,4}     43: {14}
    14: {1,1}   29: {10}      44: {1,5}
    15: {2,2}   30: {1,1,1}   45: {2,3}
		

Crossrefs

Indices of empty and singleton rows are A000961.

Row lengths are A001221.

Depends only on rootless base A052410, see A007916.

Row minima are A055396.

Rows have A071625 distinct elements.

Indices of constant rows are A072774.

Indices of strict rows are A130091.

Rows have Heinz numbers A367580.

Row sums are A367581.

Row maxima are A367583, opposite A367587.

Index of first row with Heinz number n is A367584.

Sorted row indices of first appearances are A367585.

Indices of rows of the form {1,1,...} are A367586.

Agrees with sorted prime signature at A367683, counted by A367682.

A submultiset of prime indices at A367685, counted by A367684.

A007947 gives squarefree kernel.

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

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

A181819 gives prime shadow, with an inverse A181821.

A238747 gives prime metasignature, reversed A353742.

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

A367582 counts partitions by sum of multiset multiplicity kernel.

Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.

Programs

Mathematica

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

Formula

For all positive integers n and k, row n^k is the same as row n.