OFFSET
1,2
COMMENTS
We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. As an operation on multisets, MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.
EXAMPLE
The terms together with their prime indices begin:
1: {} 28: {1,1,4} 60: {1,1,2,3}
2: {1} 29: {10} 61: {18}
3: {2} 30: {1,2,3} 63: {2,2,4}
5: {3} 31: {11} 67: {19}
6: {1,2} 35: {3,4} 68: {1,1,7}
7: {4} 37: {12} 71: {20}
11: {5} 41: {13} 73: {21}
12: {1,1,2} 43: {14} 76: {1,1,8}
13: {6} 44: {1,1,5} 77: {4,5}
15: {2,3} 45: {2,2,3} 79: {22}
17: {7} 47: {15} 83: {23}
19: {8} 52: {1,1,6} 89: {24}
20: {1,1,3} 53: {16} 90: {1,2,2,3}
23: {9} 59: {17} 92: {1,1,9}
MATHEMATICA
nn=100;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n, nn}];
Select[Range[nn], FreeQ[Take[qq, #-1], qq[[#]]]&]
CROSSREFS
All terms are rootless A007916 (have no positive integer roots).
Positions of squarefree terms appear to be A073485.
Contains no nonprime prime powers A246547.
Sorted positions of first appearances in A367580.
Sorted version of A367584.
Complement of A367768.
A007947 gives squarefree kernel.
A071625 counts distinct prime exponents.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 29 2023
STATUS
approved