OFFSET
1,2
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.
EXAMPLE
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)
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<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], aperQ[aug[primeMS[#]//Reverse]]&]
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 09 2019
STATUS
approved