A367685 Numbers divisible by their multiset multiplicity kernel.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107

Offset: 1

Author

Gus Wiseman, Nov 30 2023

Keywords

• nonn

Comments

First differs from A344586 in lacking 120.

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.

First differs from A212165 at n=73: A212165(73)=120 is not a term of this. - Amiram Eldar, Dec 04 2023

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}
		

Crossrefs

Includes all prime-powers A000961.

The only squarefree terms are the primes A008578.

Partitions of this type are counted by A367684.

A007947 gives squarefree kernel.

A027746 lists prime factors, length A001222, indices A112798.

A027748 lists distinct prime factors, length A001221, indices A304038.

A071625 counts distinct prime exponents.

A124010 gives multiset of multiplicities (prime signature), sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A367579 lists MMK, ranks A367580, sum A367581, max A367583.

Cf. A005117, A039956, A051904, A052409, A072774, A130091, A367582, A367584, A367586, A367682.

Programs

Mathematica

mmk[n_Integer]:= Product[Min[#]^Length[#]&[First/@Select[FactorInteger[n], Last[#]==k&]], {k,Union[Last/@FactorInteger[n]]}];
Select[Range[100], Divisible[#,mmk[#]]&]