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 cokernel MMC(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 max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}.
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: {5,5}
4: {1} 19: {8} 34: {7,7}
5: {3} 20: {1,3} 35: {4,4}
6: {2,2} 21: {4,4} 36: {2,2}
7: {4} 22: {5,5} 37: {12}
8: {1} 23: {9} 38: {8,8}
9: {2} 24: {1,2} 39: {6,6}
10: {3,3} 25: {3} 40: {1,3}
11: {5} 26: {6,6} 41: {13}
12: {1,2} 27: {2} 42: {4,4,4}
13: {6} 28: {1,4} 43: {14}
14: {4,4} 29: {10} 44: {1,5}
15: {3,3} 30: {3,3,3} 45: {2,3}
MATHEMATICA
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[mmc[PrimePi /@ Join@@ConstantArray@@@If[n==1, {}, FactorInteger[n]]], {n, 100}]
CROSSREFS
Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Row maxima are A061395.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Row minima are A367587.
Rows have Heinz numbers A367859.
Row sums are A367860.
A007947 gives squarefree kernel.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Dec 03 2023
STATUS
approved