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

%I A329133 #6 Nov 09 2019 16:25:58
%S A329133 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,
%T A329133 30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A329133 54,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74
%N A329133 Numbers whose augmented differences of prime indices are an aperiodic sequence.
%C A329133 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).
%C A329133 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.
%C A329133 A finite sequence is aperiodic if its cyclic rotations are all different.
%e A329133 The sequence of terms together with their augmented differences of prime indices begins:
%e A329133     1: ()
%e A329133     2: (1)
%e A329133     3: (2)
%e A329133     5: (3)
%e A329133     6: (2,1)
%e A329133     7: (4)
%e A329133     9: (1,2)
%e A329133    10: (3,1)
%e A329133    11: (5)
%e A329133    12: (2,1,1)
%e A329133    13: (6)
%e A329133    14: (4,1)
%e A329133    17: (7)
%e A329133    18: (1,2,1)
%e A329133    19: (8)
%e A329133    20: (3,1,1)
%e A329133    21: (3,2)
%e A329133    22: (5,1)
%e A329133    23: (9)
%e A329133    24: (2,1,1,1)
%t A329133 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A329133 aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
%t A329133 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A329133 Select[Range[100],aperQ[aug[primeMS[#]//Reverse]]&]
%Y A329133 Complement of A329132.
%Y A329133 These are the Heinz numbers of the partitions counted by A329136.
%Y A329133 Aperiodic binary words are A027375.
%Y A329133 Aperiodic compositions are A000740.
%Y A329133 Numbers whose binary expansion is aperiodic are A328594.
%Y A329133 Numbers whose prime signature is aperiodic are A329139.
%Y A329133 Numbers whose differences of prime indices are aperiodic are A329135.
%Y A329133 Cf. A056239, A112798, A121016, A124010, A152061, A246029, A325351, A325389, A329134, A329140.
%K A329133 nonn
%O A329133 1,2
%A A329133 _Gus Wiseman_, Nov 09 2019