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A367861
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Numbers k whose multiset multiplicity cokernel (in which each prime exponent becomes the greatest prime factor with that exponent) is different from that of all positive integers less than k.
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1
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1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 26, 28, 29, 30, 31, 34, 37, 38, 41, 42, 43, 44, 45, 46, 47, 52, 53, 58, 59, 60, 61, 62, 63, 66, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 84, 86, 89, 90, 92, 94, 97, 99, 101, 102, 103, 106, 107, 109, 113
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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 cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {} 23: {9} 47: {15}
2: {1} 26: {1,6} 52: {1,1,6}
3: {2} 28: {1,1,4} 53: {16}
5: {3} 29: {10} 58: {1,10}
6: {1,2} 30: {1,2,3} 59: {17}
7: {4} 31: {11} 60: {1,1,2,3}
10: {1,3} 34: {1,7} 61: {18}
11: {5} 37: {12} 62: {1,11}
12: {1,1,2} 38: {1,8} 63: {2,2,4}
13: {6} 41: {13} 66: {1,2,5}
14: {1,4} 42: {1,2,4} 67: {19}
17: {7} 43: {14} 68: {1,1,7}
19: {8} 44: {1,1,5} 71: {20}
20: {1,1,3} 45: {2,2,3} 73: {21}
22: {1,5} 46: {1,9} 74: {1,12}
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MATHEMATICA
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nn=100;
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q, Count[q, #]==i&], {i, mts}]]];
qq=Table[Times@@mmc[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).
For kernel instead of cokernel we have A367585, sorted version of A367584.
Sorted positions of first appearances in A367859.
A071625 counts distinct prime exponents.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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