A334298 Numbers whose prime signature is a reversed Lyndon word.

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

Author

Gus Wiseman, Jun 10 2020

Keywords

• nonn

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.

Examples

			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}
		

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.

Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.

Programs

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[#]]]]&]