A329135 Numbers whose differences of prime indices are an aperiodic word.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Author

Gus Wiseman, Nov 09 2019

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A sequence is aperiodic if its cyclic rotations are all different.

Examples

			The sequence of terms together with their differences of prime indices begins:
    1: ()
    2: ()
    3: ()
    4: (0)
    5: ()
    6: (1)
    7: ()
    9: (0)
   10: (2)
   11: ()
   12: (0,1)
   13: ()
   14: (3)
   15: (1)
   17: ()
   18: (1,0)
   19: ()
   20: (0,2)
   21: (2)
   22: (4)
		

Crossrefs

Complement of A329134.

These are the Heinz numbers of the partitions counted by A329137.

Aperiodic compositions are A000740.

Aperiodic binary words are A027375.

Numbers whose binary expansion is aperiodic are A328594.

Numbers whose prime signature is aperiodic are A329139.

Cf. A056239, A112798, A124010, A152061, A329133, A329136, A329140.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Differences[primeMS[#]]]&]