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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.
10

%I #8 Apr 28 2024 16:18:15

%S 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 52,53,59,60,61,63,67,68,71,73,76,77,79,83,89,90,92,97,99,101,103,105,

%U 107,109,113,116,117,124,127,131,137,139,140,143,148,149,150

%N 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 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 The terms together with their prime indices begin:

%e 1: {} 28: {1,1,4} 60: {1,1,2,3}

%e 2: {1} 29: {10} 61: {18}

%e 3: {2} 30: {1,2,3} 63: {2,2,4}

%e 5: {3} 31: {11} 67: {19}

%e 6: {1,2} 35: {3,4} 68: {1,1,7}

%e 7: {4} 37: {12} 71: {20}

%e 11: {5} 41: {13} 73: {21}

%e 12: {1,1,2} 43: {14} 76: {1,1,8}

%e 13: {6} 44: {1,1,5} 77: {4,5}

%e 15: {2,3} 45: {2,2,3} 79: {22}

%e 17: {7} 47: {15} 83: {23}

%e 19: {8} 52: {1,1,6} 89: {24}

%e 20: {1,1,3} 53: {16} 90: {1,2,2,3}

%e 23: {9} 59: {17} 92: {1,1,9}

%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], FreeQ[Take[qq,#-1], qq[[#]]]&]

%Y Contains all primes A000040 but no other perfect powers A001597.

%Y All terms are rootless A007916 (have no positive integer roots).

%Y Positions of squarefree terms appear to be A073485.

%Y Contains no nonprime prime powers A246547.

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

%Y Sorted positions of first appearances in A367580.

%Y Sorted version of A367584.

%Y Complement of A367768.

%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. A020639, A051904, A072774, A130091, A181819, A238747, A367582, A367685.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 29 2023