OFFSET
1,2
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}.
FORMULA
For all positive integers n and k, row n^k is the same as row n.
EXAMPLE
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}
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}]
CROSSREFS
Indices of empty and singleton rows are A000961.
Row lengths are A001221.
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.
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.
A007947 gives squarefree kernel.
A367582 counts partitions by sum of multiset multiplicity kernel.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Nov 25 2023
STATUS
approved