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%I A329142 #7 Nov 10 2019 20:29:05
%S A329142 1,12,20,24,28,40,44,45,48,52,56,60,63,68,72,76,80,84,88,90,92,96,99,
%T A329142 104,112,116,117,120,124,126,132,135,136,140,144,148,152,153,156,160,
%U A329142 164,168,171,172,175,176,180,184,188,189,192,198,200,204,207,208,212
%N A329142 Numbers whose prime signature is not a necklace.
%C A329142 After a(1) = 1, first differs from A112769 in lacking 1350.
%C A329142 A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
%C A329142 A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.
%e A329142 The sequence of terms together with their prime signatures begins:
%e A329142    1: ()
%e A329142   12: (2,1)
%e A329142   20: (2,1)
%e A329142   24: (3,1)
%e A329142   28: (2,1)
%e A329142   40: (3,1)
%e A329142   44: (2,1)
%e A329142   45: (2,1)
%e A329142   48: (4,1)
%e A329142   52: (2,1)
%e A329142   56: (3,1)
%e A329142   60: (2,1,1)
%e A329142   63: (2,1)
%e A329142   68: (2,1)
%e A329142   72: (3,2)
%e A329142   76: (2,1)
%e A329142   80: (4,1)
%e A329142   84: (2,1,1)
%e A329142   88: (3,1)
%e A329142   90: (1,2,1)
%e A329142   92: (2,1)
%t A329142 neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A329142 Select[Range[100],#==1||!neckQ[Last/@FactorInteger[#]]&]
%Y A329142 Complement of A329138.
%Y A329142 Binary necklaces are A000031.
%Y A329142 Non-necklace compositions are A329145.
%Y A329142 Numbers whose reversed binary expansion is a necklace are A328595.
%Y A329142 Numbers whose prime signature is a Lyndon word are A329131.
%Y A329142 Numbers whose prime signature is periodic are A329140.
%Y A329142 Cf. A001037, A008965, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329139.
%K A329142 nonn
%O A329142 1,2
%A A329142 _Gus Wiseman_, Nov 09 2019

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