%I #6 Nov 10 2019 20:29:34

%S 2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,25,26,27,29,30,

%T 31,32,33,34,35,36,37,38,39,41,42,43,46,47,49,50,51,53,54,55,57,58,59,

%U 61,62,64,65,66,67,69,70,71,73,74,75,77,78,79,81,82,83

%N Numbers whose prime signature is a necklace.

%C First differs from A304678 in having 1350 = 2^1 * 3^3 * 5^2. First differs from A316529 in having 150 = 2^1 * 3^1 * 5^2.

%C A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.

%C A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

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

%e 2: (1)

%e 3: (1)

%e 4: (2)

%e 5: (1)

%e 6: (1,1)

%e 7: (1)

%e 8: (3)

%e 9: (2)

%e 10: (1,1)

%e 11: (1)

%e 13: (1)

%e 14: (1,1)

%e 15: (1,1)

%e 16: (4)

%e 17: (1)

%e 18: (1,2)

%e 19: (1)

%e 21: (1,1)

%e 22: (1,1)

%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];

%t Select[Range[2,100],neckQ[Last/@FactorInteger[#]]&]

%Y Complement of A329142.

%Y Binary necklaces are A000031.

%Y Necklace compositions are A008965.

%Y Numbers whose reversed binary expansion is a necklace are A328595.

%Y Numbers whose prime signature is a Lyndon word are A329131.

%Y Numbers whose prime signature is aperiodic are A329139.

%Y Cf. A001037, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329140.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 09 2019