A329142 Numbers whose prime signature is not a necklace.

1, 12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212

Offset: 1

Author

Gus Wiseman, Nov 09 2019

Keywords

• nonn

Comments

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

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

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

Examples

			The sequence of terms together with their prime signatures begins:
   1: ()
  12: (2,1)
  20: (2,1)
  24: (3,1)
  28: (2,1)
  40: (3,1)
  44: (2,1)
  45: (2,1)
  48: (4,1)
  52: (2,1)
  56: (3,1)
  60: (2,1,1)
  63: (2,1)
  68: (2,1)
  72: (3,2)
  76: (2,1)
  80: (4,1)
  84: (2,1,1)
  88: (3,1)
  90: (1,2,1)
  92: (2,1)
		

Crossrefs

Complement of A329138.

Binary necklaces are A000031.

Non-necklace compositions are A329145.

Numbers whose reversed binary expansion is a necklace are A328595.

Numbers whose prime signature is a Lyndon word are A329131.

Numbers whose prime signature is periodic are A329140.

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

Programs

Mathematica

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],#==1||!neckQ[Last/@FactorInteger[#]]&]