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A334298
Numbers whose prime signature is a reversed Lyndon word.
2
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
OFFSET
1,2
COMMENTS
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.
EXAMPLE
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}
MATHEMATICA
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[#]]]]&]
CROSSREFS
The non-reversed version is A329131.
Lyndon compositions are A059966.
Prime signature is A124010.
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.
Sequence in context: A268375 A048683 A231876 * A364160 A304686 A354921
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 10 2020
STATUS
approved