A329142 - OEIS

%I #7 Nov 10 2019 20:29:05

%S 1,12,20,24,28,40,44,45,48,52,56,60,63,68,72,76,80,84,88,90,92,96,99,

%T 104,112,116,117,120,124,126,132,135,136,140,144,148,152,153,156,160,

%U 164,168,171,172,175,176,180,184,188,189,192,198,200,204,207,208,212

%N Numbers whose prime signature is not a necklace.

%C After a(1) = 1, first differs from A112769 in lacking 1350.

%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    1: ()

%e   12: (2,1)

%e   20: (2,1)

%e   24: (3,1)

%e   28: (2,1)

%e   40: (3,1)

%e   44: (2,1)

%e   45: (2,1)

%e   48: (4,1)

%e   52: (2,1)

%e   56: (3,1)

%e   60: (2,1,1)

%e   63: (2,1)

%e   68: (2,1)

%e   72: (3,2)

%e   76: (2,1)

%e   80: (4,1)

%e   84: (2,1,1)

%e   88: (3,1)

%e   90: (1,2,1)

%e   92: (2,1)

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

%t Select[Range[100],#==1||!neckQ[Last/@FactorInteger[#]]&]

%Y Complement of A329138.

%Y Binary necklaces are A000031.

%Y Non-necklace compositions are A329145.

%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 periodic are A329140.

%Y Cf. A001037, A008965, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329139.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 09 2019