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A367768 Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.

%I A367768 #7 Dec 01 2023 23:26:28
%S A367768 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 A367768 50,51,54,55,56,57,58,62,64,65,66,69,70,72,74,75,78,80,81,82,84,85,86,
%U A367768 87,88,91,93,94,95,96,98,100,102,104,106,108,110,111,112,114
%N A367768 Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.
%C A367768 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 A367768 A367580(a(k)) = A367580(i) for some i < a(k).
%e A367768 The terms together with their prime indices begin:
%e A367768     4: {1,1}
%e A367768     8: {1,1,1}
%e A367768     9: {2,2}
%e A367768    10: {1,3}
%e A367768    14: {1,4}
%e A367768    16: {1,1,1,1}
%e A367768    18: {1,2,2}
%e A367768    21: {2,4}
%e A367768    22: {1,5}
%e A367768    24: {1,1,1,2}
%e A367768    25: {3,3}
%e A367768    26: {1,6}
%e A367768    27: {2,2,2}
%e A367768    32: {1,1,1,1,1}
%e A367768    33: {2,5}
%e A367768    34: {1,7}
%e A367768    36: {1,1,2,2}
%t A367768 nn=100;
%t A367768 mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367768 qq=Table[Times@@mmk[Join @@ ConstantArray@@@FactorInteger[n]],{n,nn}];
%t A367768 Select[Range[nn], MemberQ[Take[qq,#-1], qq[[#]]]&]
%Y A367768 The squarefree case is A073486, complement A073485.
%Y A367768 The MMK triangle is A367579, sum A367581, min A055396, max A367583.
%Y A367768 Sorted positions of non-first appearances in A367580.
%Y A367768 The complement is A367585, sorted version of A367584.
%Y A367768 A007947 gives squarefree kernel.
%Y A367768 A027746 lists prime factors, length A001222, indices A112798.
%Y A367768 A027748 lists distinct prime factors, length A001221, indices A304038.
%Y A367768 A071625 counts distinct prime exponents.
%Y A367768 A124010 gives prime signature, sorted A118914.
%Y A367768 Cf. A051904, A072774, A130091, A181819, A246547, A367685.
%K A367768 nonn
%O A367768 1,1
%A A367768 _Gus Wiseman_, Dec 01 2023