A367685 - OEIS

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

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OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

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}

MATHEMATICA

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

Select[Range[100], Divisible[#, mmk[#]]&]

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.

Sequence in context: A371782 A363597 A212165 * A344586 A319161 A325370

Adjacent sequences: A367682 A367683 A367684 * A367686 A367687 A367688

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 30 2023

STATUS

approved