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A367585 Numbers k whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is different from that of all positive integers less than k.

%I A367585 #8 Apr 28 2024 16:18:15
%S A367585 1,2,3,5,6,7,11,12,13,15,17,19,20,23,28,29,30,31,35,37,41,43,44,45,47,
%T A367585 52,53,59,60,61,63,67,68,71,73,76,77,79,83,89,90,92,97,99,101,103,105,
%U A367585 107,109,113,116,117,124,127,131,137,139,140,143,148,149,150
%N A367585 Numbers k whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is different from that of all positive integers less than k.
%C A367585 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.
%e A367585 The terms together with their prime indices begin:
%e A367585      1: {}         28: {1,1,4}    60: {1,1,2,3}
%e A367585      2: {1}        29: {10}       61: {18}
%e A367585      3: {2}        30: {1,2,3}    63: {2,2,4}
%e A367585      5: {3}        31: {11}       67: {19}
%e A367585      6: {1,2}      35: {3,4}      68: {1,1,7}
%e A367585      7: {4}        37: {12}       71: {20}
%e A367585     11: {5}        41: {13}       73: {21}
%e A367585     12: {1,1,2}    43: {14}       76: {1,1,8}
%e A367585     13: {6}        44: {1,1,5}    77: {4,5}
%e A367585     15: {2,3}      45: {2,2,3}    79: {22}
%e A367585     17: {7}        47: {15}       83: {23}
%e A367585     19: {8}        52: {1,1,6}    89: {24}
%e A367585     20: {1,1,3}    53: {16}       90: {1,2,2,3}
%e A367585     23: {9}        59: {17}       92: {1,1,9}
%t A367585 nn=100;
%t A367585 mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367585 qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,nn}];
%t A367585 Select[Range[nn], FreeQ[Take[qq,#-1], qq[[#]]]&]
%Y A367585 Contains all primes A000040 but no other perfect powers A001597.
%Y A367585 All terms are rootless A007916 (have no positive integer roots).
%Y A367585 Positions of squarefree terms appear to be A073485.
%Y A367585 Contains no nonprime prime powers A246547.
%Y A367585 The MMK triangle is A367579, sum A367581, min A055396, max A367583.
%Y A367585 Sorted positions of first appearances in A367580.
%Y A367585 Sorted version of A367584.
%Y A367585 Complement of A367768.
%Y A367585 A007947 gives squarefree kernel.
%Y A367585 A027746 lists prime factors, length A001222, indices A112798.
%Y A367585 A027748 lists distinct prime factors, length A001221, indices A304038.
%Y A367585 A071625 counts distinct prime exponents.
%Y A367585 A124010 gives prime signature, sorted A118914.
%Y A367585 Cf. A020639, A051904, A072774, A130091, A181819, A238747, A367582, A367685.
%K A367585 nonn
%O A367585 1,2
%A A367585 _Gus Wiseman_, Nov 29 2023