cp's OEIS Frontend

A329132 Numbers whose augmented differences of prime indices are a periodic sequence.

%I A329132 #5 Nov 09 2019 16:25:51
%S A329132 4,8,15,16,32,55,64,90,105,119,128,225,253,256,403,512,540,550,697,
%T A329132 893,935,1024,1155,1350,1357,1666,1943,2048,2263,3025,3071,3150,3240,
%U A329132 3375,3451,3927,3977,4096,4429,5123,5500,5566,6731,7735,8083,8100,8192,9089
%N A329132 Numbers whose augmented differences of prime indices are a periodic sequence.
%C A329132 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 A329132 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 A329132 A sequence is periodic if its cyclic rotations are not all different.
%e A329132 The sequence of terms together with their augmented differences of prime indices begins:
%e A329132      4: (1,1)
%e A329132      8: (1,1,1)
%e A329132     15: (2,2)
%e A329132     16: (1,1,1,1)
%e A329132     32: (1,1,1,1,1)
%e A329132     55: (3,3)
%e A329132     64: (1,1,1,1,1,1)
%e A329132     90: (2,1,2,1)
%e A329132    105: (2,2,2)
%e A329132    119: (4,4)
%e A329132    128: (1,1,1,1,1,1,1)
%e A329132    225: (1,2,1,2)
%e A329132    253: (5,5)
%e A329132    256: (1,1,1,1,1,1,1,1)
%e A329132    403: (6,6)
%e A329132    512: (1,1,1,1,1,1,1,1,1)
%e A329132    540: (2,1,1,2,1,1)
%e A329132    550: (3,1,3,1)
%e A329132    697: (7,7)
%e A329132    893: (8,8)
%t A329132 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A329132 aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
%t A329132 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A329132 Select[Range[100],!aperQ[aug[primeMS[#]//Reverse]]&]
%Y A329132 Complement of A329133.
%Y A329132 These are the Heinz numbers of the partitions counted by A329143.
%Y A329132 Periodic binary words are A152061.
%Y A329132 Periodic compositions are A178472.
%Y A329132 Numbers whose binary expansion is periodic are A121016.
%Y A329132 Numbers whose prime signature is periodic are A329140.
%Y A329132 Numbers whose differences of prime indices are periodic are A329134.
%Y A329132 Cf. A000961, A027375, A056239, A112798, A325356, A325389, A325394, A328594, A329135, A329136, A329139.
%K A329132 nonn
%O A329132 1,1
%A A329132 _Gus Wiseman_, Nov 06 2019