A072609 - OEIS

%I #8 Dec 21 2014 19:36:56

%S 0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,

%U 0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range [...1111..]. AC-range corresponds to 0, while DC-range labeled by 1.

%F a(n)=Mod[A004648(n), 2]*Mod[A004648(n+1), 2]= A072608(n)*A072608(n+1)

%e Take n = 11,12,13,14: A004648[n]=9,1,2,1. Parity A072608(n) = 1,1,0,1. So ..11.. transforms into 01 between n = 11 and n = 12: a(11) = 1, a(12)=0. With increasing n, A072609(n) changes from ..0000.. into ...1111. reflected by this sequence. by a range consisting only of 1-s. This secondary alternation also goes on.

%t mm[x_] := Mod[Mod[Prime[x], x], 2] Table[mm[w]*mm[w+1], {w, 1, 256}]

%t Times@@@Partition[Table[Mod[Mod[Prime[n],n],2],{n,110}],2,1] (* _Harvey P. Dale_, Dec 21 2014 *)

%Y Cf. A004648, A072608.

%K nice,nonn

%O 1,1

%A _Labos Elemer_, Jun 24 2002