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

%I A329135 #5 Nov 09 2019 16:26:13
%S A329135 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,
%T A329135 29,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A329135 54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73
%N A329135 Numbers whose differences of prime indices are an aperiodic word.
%C A329135 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 A329135 A sequence is aperiodic if its cyclic rotations are all different.
%e A329135 The sequence of terms together with their differences of prime indices begins:
%e A329135     1: ()
%e A329135     2: ()
%e A329135     3: ()
%e A329135     4: (0)
%e A329135     5: ()
%e A329135     6: (1)
%e A329135     7: ()
%e A329135     9: (0)
%e A329135    10: (2)
%e A329135    11: ()
%e A329135    12: (0,1)
%e A329135    13: ()
%e A329135    14: (3)
%e A329135    15: (1)
%e A329135    17: ()
%e A329135    18: (1,0)
%e A329135    19: ()
%e A329135    20: (0,2)
%e A329135    21: (2)
%e A329135    22: (4)
%t A329135 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A329135 aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
%t A329135 Select[Range[100],aperQ[Differences[primeMS[#]]]&]
%Y A329135 Complement of A329134.
%Y A329135 These are the Heinz numbers of the partitions counted by A329137.
%Y A329135 Aperiodic compositions are A000740.
%Y A329135 Aperiodic binary words are A027375.
%Y A329135 Numbers whose binary expansion is aperiodic are A328594.
%Y A329135 Numbers whose prime signature is aperiodic are A329139.
%Y A329135 Cf. A056239, A112798, A124010, A152061, A329133, A329136, A329140.
%K A329135 nonn
%O A329135 1,2
%A A329135 _Gus Wiseman_, Nov 09 2019