A329133 Numbers whose augmented differences of prime indices are an aperiodic sequence.

1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Author

Gus Wiseman, Nov 09 2019

Keywords

• nonn

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

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

Examples

			The sequence of terms together with their augmented differences of prime indices begins:
    1: ()
    2: (1)
    3: (2)
    5: (3)
    6: (2,1)
    7: (4)
    9: (1,2)
   10: (3,1)
   11: (5)
   12: (2,1,1)
   13: (6)
   14: (4,1)
   17: (7)
   18: (1,2,1)
   19: (8)
   20: (3,1,1)
   21: (3,2)
   22: (5,1)
   23: (9)
   24: (2,1,1,1)
		

Crossrefs

Complement of A329132.

These are the Heinz numbers of the partitions counted by A329136.

Aperiodic binary words are A027375.

Aperiodic compositions are A000740.

Numbers whose binary expansion is aperiodic are A328594.

Numbers whose prime signature is aperiodic are A329139.

Numbers whose differences of prime indices are aperiodic are A329135.

Cf. A056239, A112798, A121016, A124010, A152061, A246029, A325351, A325389, A329134, 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];
aug[y_]:=Table[If[i