A329131 Numbers whose prime signature is a Lyndon word.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 150, 151, 157, 162, 163, 167

Offset: 1

Author

Gus Wiseman, Nov 06 2019

Keywords

• nonn

Comments

First differs from A133811 in having 50.

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 30870 is (1,2,1,3), which is a Lyndon word, so 30870 is in the sequence.
The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
		

Crossrefs

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.

Lyndon compositions are A059966.

Prime signature is A124010.

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

Programs

Mathematica

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
Select[Range[2,100],lynQ[Last/@FactorInteger[#]]&]

Formula

Intersection of A329138 and A329139.