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A367585
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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.
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10
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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, 52, 53, 59, 60, 61, 63, 67, 68, 71, 73, 76, 77, 79, 83, 89, 90, 92, 97, 99, 101, 103, 105, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 140, 143, 148, 149, 150
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {} 28: {1,1,4} 60: {1,1,2,3}
2: {1} 29: {10} 61: {18}
3: {2} 30: {1,2,3} 63: {2,2,4}
5: {3} 31: {11} 67: {19}
6: {1,2} 35: {3,4} 68: {1,1,7}
7: {4} 37: {12} 71: {20}
11: {5} 41: {13} 73: {21}
12: {1,1,2} 43: {14} 76: {1,1,8}
13: {6} 44: {1,1,5} 77: {4,5}
15: {2,3} 45: {2,2,3} 79: {22}
17: {7} 47: {15} 83: {23}
19: {8} 52: {1,1,6} 89: {24}
20: {1,1,3} 53: {16} 90: {1,2,2,3}
23: {9} 59: {17} 92: {1,1,9}
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MATHEMATICA
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nn=100;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n, nn}];
Select[Range[nn], FreeQ[Take[qq, #-1], qq[[#]]]&]
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CROSSREFS
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All terms are rootless A007916 (have no positive integer roots).
Positions of squarefree terms appear to be A073485.
Contains no nonprime prime powers A246547.
Sorted positions of first appearances in A367580.
A071625 counts distinct prime exponents.
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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