A329134 - OEIS

%I #5 Nov 09 2019 16:26:06

%S 8,16,27,30,32,64,81,105,110,125,128,180,210,238,243,256,273,343,385,

%T 450,506,512,625,627,729,806,935,1001,1024,1080,1100,1131,1155,1331,

%U 1394,1495,1575,1729,1786,1870,1887,2048,2187,2197,2310,2401,2431,2451,2635

%N Numbers whose differences of prime indices are a periodic word.

%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 differences of prime indices begins:

%e      8: (0,0)

%e     16: (0,0,0)

%e     27: (0,0)

%e     30: (1,1)

%e     32: (0,0,0,0)

%e     64: (0,0,0,0,0)

%e     81: (0,0,0)

%e    105: (1,1)

%e    110: (2,2)

%e    125: (0,0)

%e    128: (0,0,0,0,0,0)

%e    180: (0,1,0,1)

%e    210: (1,1,1)

%e    238: (3,3)

%e    243: (0,0,0,0)

%e    256: (0,0,0,0,0,0,0)

%e    273: (2,2)

%e    343: (0,0)

%e    385: (1,1)

%e    450: (1,0,1,0)

%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 Select[Range[10000],!aperQ[Differences[primeMS[#]]]&]

%Y Complement of A329135.

%Y These are the Heinz numbers of the partitions counted by A329144.

%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 Cf. A000740, A027375, A056239, A112798, A124010, A328594, A329132, A329139.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 09 2019