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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 20 X 20 color coded list of terms, color function: gray = 1, red = prime, gold = composite prime power, green = squarefree composite, blue-purple = numbers neither squarefree nor prime powers. Bright green = primorial, light green = even squarefree semiprime, light blue = highly composite, middle blue = in A055932, purple = squareful but not a prime power.
Michael De Vlieger, 485 X 485 = 235225-term raster with the same color code as above.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
9: {2,2}
10: {1,3}
12: {1,1,2}
14: {1,4}
18: {1,2,2}
22: {1,5}
26: {1,6}
30: {1,2,3}
34: {1,7}
38: {1,8}
40: {1,1,1,3}
42: {1,2,4}
46: {1,9}
58: {1,10}
62: {1,11}
66: {1,2,5}
70: {1,3,4}
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Select[Range[100], #==1||Sort[Last/@FactorInteger[#]] == mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]]&]
CROSSREFS
Squarefree terms are A039956.
Prime-power terms are A307539.
Partitions of this type are counted by A367682.
A007947 gives squarefree kernel.
A367582 counts partitions by sum of multiset multiplicity kernel.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2023
STATUS
approved