A367768 - OEIS

%I #7 Dec 01 2023 23:26:28

%S 4,8,9,10,14,16,18,21,22,24,25,26,27,32,33,34,36,38,39,40,42,46,48,49,

%T 50,51,54,55,56,57,58,62,64,65,66,69,70,72,74,75,78,80,81,82,84,85,86,

%U 87,88,91,93,94,95,96,98,100,102,104,106,108,110,111,112,114

%N Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.

%C 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.

%F A367580(a(k)) = A367580(i) for some i < a(k).

%e The terms together with their prime indices begin:

%e     4: {1,1}

%e     8: {1,1,1}

%e     9: {2,2}

%e    10: {1,3}

%e    14: {1,4}

%e    16: {1,1,1,1}

%e    18: {1,2,2}

%e    21: {2,4}

%e    22: {1,5}

%e    24: {1,1,1,2}

%e    25: {3,3}

%e    26: {1,6}

%e    27: {2,2,2}

%e    32: {1,1,1,1,1}

%e    33: {2,5}

%e    34: {1,7}

%e    36: {1,1,2,2}

%t nn=100;

%t mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];

%t qq=Table[Times@@mmk[Join @@ ConstantArray@@@FactorInteger[n]],{n,nn}];

%t Select[Range[nn], MemberQ[Take[qq,#-1], qq[[#]]]&]

%Y The squarefree case is A073486, complement A073485.

%Y The MMK triangle is A367579, sum A367581, min A055396, max A367583.

%Y Sorted positions of non-first appearances in A367580.

%Y The complement is A367585, sorted version of A367584.

%Y A007947 gives squarefree kernel.

%Y A027746 lists prime factors, length A001222, indices A112798.

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

%Y A071625 counts distinct prime exponents.

%Y A124010 gives prime signature, sorted A118914.

%Y Cf. A051904, A072774, A130091, A181819, A246547, A367685.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 01 2023