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A334298
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Numbers whose prime signature is a reversed Lyndon word.
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2
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1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116
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OFFSET
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1,2
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COMMENTS
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A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.
A number's prime signature is the sequence of positive exponents in its prime factorization.
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LINKS
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EXAMPLE
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The prime signature of 4200 is (3,1,2,1), which is a reversed Lyndon word, so 4200 is in the sequence.
The sequence of terms together with their prime indices begins:
1: {} 23: {9} 48: {1,1,1,1,2}
2: {1} 24: {1,1,1,2} 49: {4,4}
3: {2} 25: {3,3} 52: {1,1,6}
4: {1,1} 27: {2,2,2} 53: {16}
5: {3} 28: {1,1,4} 56: {1,1,1,4}
7: {4} 29: {10} 59: {17}
8: {1,1,1} 31: {11} 60: {1,1,2,3}
9: {2,2} 32: {1,1,1,1,1} 61: {18}
11: {5} 37: {12} 63: {2,2,4}
12: {1,1,2} 40: {1,1,1,3} 64: {1,1,1,1,1,1}
13: {6} 41: {13} 67: {19}
16: {1,1,1,1} 43: {14} 68: {1,1,7}
17: {7} 44: {1,1,5} 71: {20}
19: {8} 45: {2,2,3} 72: {1,1,1,2,2}
20: {1,1,3} 47: {15} 73: {21}
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MATHEMATICA
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lynQ[q_]:=Length[q]==0||Array[Union[{q, RotateRight[q, #1]}]=={q, RotateRight[q, #1]}&, Length[q]-1, 1, And];
Select[Range[100], lynQ[Reverse[Last/@If[#==1, {}, FactorInteger[#]]]]&]
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CROSSREFS
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The non-reversed version is A329131.
Numbers with strictly decreasing prime multiplicities are A304686.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose prime signature is a necklace are A329138.
Numbers whose prime signature is aperiodic are A329139.
Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.
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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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