A334298 - OEIS

A334298

Numbers whose prime signature is a reversed Lyndon word.

%I #4 Jun 10 2020 14:06:17

%S 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,

%T 41,43,44,45,47,48,49,52,53,56,59,60,61,63,64,67,68,71,72,73,76,79,80,

%U 81,83,84,88,89,92,96,97,99,101,103,104,107,109,112,113,116

%N Numbers whose prime signature is a reversed Lyndon word.

%C A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.

%C A number's prime signature is the sequence of positive exponents in its prime factorization.

%e The prime signature of 4200 is (3,1,2,1), which is a reversed Lyndon word, so 4200 is in the sequence.

%e The sequence of terms together with their prime indices begins:

%e 1: {} 23: {9} 48: {1,1,1,1,2}

%e 2: {1} 24: {1,1,1,2} 49: {4,4}

%e 3: {2} 25: {3,3} 52: {1,1,6}

%e 4: {1,1} 27: {2,2,2} 53: {16}

%e 5: {3} 28: {1,1,4} 56: {1,1,1,4}

%e 7: {4} 29: {10} 59: {17}

%e 8: {1,1,1} 31: {11} 60: {1,1,2,3}

%e 9: {2,2} 32: {1,1,1,1,1} 61: {18}

%e 11: {5} 37: {12} 63: {2,2,4}

%e 12: {1,1,2} 40: {1,1,1,3} 64: {1,1,1,1,1,1}

%e 13: {6} 41: {13} 67: {19}

%e 16: {1,1,1,1} 43: {14} 68: {1,1,7}

%e 17: {7} 44: {1,1,5} 71: {20}

%e 19: {8} 45: {2,2,3} 72: {1,1,1,2,2}

%e 20: {1,1,3} 47: {15} 73: {21}

%t lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];

%t Select[Range[100],lynQ[Reverse[Last/@If[#==1,{},FactorInteger[#]]]]&]

%Y The non-reversed version is A329131.

%Y Lyndon compositions are A059966.

%Y Prime signature is A124010.

%Y Numbers with strictly decreasing prime multiplicities are A304686.

%Y Numbers whose reversed binary expansion is Lyndon are A328596.

%Y Numbers whose prime signature is a necklace are A329138.

%Y Numbers whose prime signature is aperiodic are A329139.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 10 2020