cp's OEIS Frontend

A367685 Numbers divisible by their multiset multiplicity kernel.

%I A367685 #13 Apr 25 2025 23:40:05
%S A367685 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,
%T A367685 37,40,41,43,44,45,47,48,49,50,52,53,54,56,59,61,63,64,67,68,71,72,73,
%U A367685 75,76,79,80,81,83,88,89,92,96,97,98,99,100,101,103,104,107
%N A367685 Numbers divisible by their multiset multiplicity kernel.
%C A367685 First differs from A344586 in lacking 120.
%C A367685 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.
%C A367685 First differs from A212165 at n=73: A212165(73)=120 is not a term of this. - _Amiram Eldar_, Dec 04 2023
%e A367685 The terms together with their prime indices begin:
%e A367685     1: {}
%e A367685     2: {1}
%e A367685     3: {2}
%e A367685     4: {1,1}
%e A367685     5: {3}
%e A367685     7: {4}
%e A367685     8: {1,1,1}
%e A367685     9: {2,2}
%e A367685    11: {5}
%e A367685    12: {1,1,2}
%e A367685    13: {6}
%e A367685    16: {1,1,1,1}
%e A367685    17: {7}
%e A367685    18: {1,2,2}
%e A367685    19: {8}
%e A367685    20: {1,1,3}
%e A367685    23: {9}
%e A367685    24: {1,1,1,2}
%t A367685 mmk[n_Integer]:= Product[Min[#]^Length[#]&[First/@Select[FactorInteger[n], Last[#]==k&]], {k,Union[Last/@FactorInteger[n]]}];
%t A367685 Select[Range[100], Divisible[#,mmk[#]]&]
%Y A367685 Includes all prime-powers A000961.
%Y A367685 The only squarefree terms are the primes A008578.
%Y A367685 Partitions of this type are counted by A367684.
%Y A367685 A007947 gives squarefree kernel.
%Y A367685 A027746 lists prime factors, length A001222, indices A112798.
%Y A367685 A027748 lists distinct prime factors, length A001221, indices A304038.
%Y A367685 A071625 counts distinct prime exponents.
%Y A367685 A124010 gives multiset of multiplicities (prime signature), sorted A118914.
%Y A367685 A181819 gives prime shadow, with an inverse A181821.
%Y A367685 A367579 lists MMK, ranks A367580, sum A367581, max A367583.
%Y A367685 Cf. A005117, A039956, A051904, A052409, A072774, A130091, A367582, A367584, A367586, A367682.
%K A367685 nonn
%O A367685 1,2
%A A367685 _Gus Wiseman_, Nov 30 2023