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Numbers whose augmented differences of prime indices are a periodic sequence.
7

%I #5 Nov 09 2019 16:25:51

%S 4,8,15,16,32,55,64,90,105,119,128,225,253,256,403,512,540,550,697,

%T 893,935,1024,1155,1350,1357,1666,1943,2048,2263,3025,3071,3150,3240,

%U 3375,3451,3927,3977,4096,4429,5123,5500,5566,6731,7735,8083,8100,8192,9089

%N Numbers whose augmented differences of prime indices are a periodic 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 sequence is periodic if its cyclic rotations are not all different.

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

%e 4: (1,1)

%e 8: (1,1,1)

%e 15: (2,2)

%e 16: (1,1,1,1)

%e 32: (1,1,1,1,1)

%e 55: (3,3)

%e 64: (1,1,1,1,1,1)

%e 90: (2,1,2,1)

%e 105: (2,2,2)

%e 119: (4,4)

%e 128: (1,1,1,1,1,1,1)

%e 225: (1,2,1,2)

%e 253: (5,5)

%e 256: (1,1,1,1,1,1,1,1)

%e 403: (6,6)

%e 512: (1,1,1,1,1,1,1,1,1)

%e 540: (2,1,1,2,1,1)

%e 550: (3,1,3,1)

%e 697: (7,7)

%e 893: (8,8)

%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 A329133.

%Y These are the Heinz numbers of the partitions counted by A329143.

%Y Periodic binary words are A152061.

%Y Periodic compositions are A178472.

%Y Numbers whose binary expansion is periodic are A121016.

%Y Numbers whose prime signature is periodic are A329140.

%Y Numbers whose differences of prime indices are periodic are A329134.

%Y Cf. A000961, A027375, A056239, A112798, A325356, A325389, A325394, A328594, A329135, A329136, A329139.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 06 2019