A329133 - OEIS

%I #6 Nov 09 2019 16:25:58

%S 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 30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,

%U 54,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74

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

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

%e The sequence of terms together with their augmented differences of prime indices begins:

%e     1: ()

%e     2: (1)

%e     3: (2)

%e     5: (3)

%e     6: (2,1)

%e     7: (4)

%e     9: (1,2)

%e    10: (3,1)

%e    11: (5)

%e    12: (2,1,1)

%e    13: (6)

%e    14: (4,1)

%e    17: (7)

%e    18: (1,2,1)

%e    19: (8)

%e    20: (3,1,1)

%e    21: (3,2)

%e    22: (5,1)

%e    23: (9)

%e    24: (2,1,1,1)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];

%t aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];

%t Select[Range[100],aperQ[aug[primeMS[#]//Reverse]]&]

%Y Complement of A329132.

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

%Y Aperiodic binary words are A027375.

%Y Aperiodic compositions are A000740.

%Y Numbers whose binary expansion is aperiodic are A328594.

%Y Numbers whose prime signature is aperiodic are A329139.

%Y Numbers whose differences of prime indices are aperiodic are A329135.

%Y Cf. A056239, A112798, A121016, A124010, A152061, A246029, A325351, A325389, A329134, A329140.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 09 2019