OFFSET
1,1
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:
4: {1,1}
8: {1,1,1}
9: {2,2}
10: {1,3}
14: {1,4}
16: {1,1,1,1}
18: {1,2,2}
21: {2,4}
22: {1,5}
24: {1,1,1,2}
25: {3,3}
26: {1,6}
27: {2,2,2}
32: {1,1,1,1,1}
33: {2,5}
34: {1,7}
36: {1,1,2,2}
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], MemberQ[Take[qq, #-1], qq[[#]]]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 01 2023
STATUS
approved