login
a(n) = (prime(n + 1) - prime(n))/2 (mod 2).
10

%I #12 Sep 01 2024 10:28:26

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

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

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

%N a(n) = (prime(n + 1) - prime(n))/2 (mod 2).

%C Questions: Is s(n) = (1/n)*Sum_{i=2..n+1} a(i) > 1/2 for all n? Does s(n) --> 1/2?

%e a(1) = (prime(2 + 1) - prime(2))/2 (mod 2) = (5 - 3)/2 (mod 2) = 1 mod 2 = 1.

%t Table[Mod[(Prime[i + 1] - Prime[i])/2, 2], {i, 2, 100}]

%t Mod[(#[[2]]-#[[1]])/2,2]&/@Partition[Prime[Range[2,110]],2,1] (* _Harvey P. Dale_, Oct 01 2018 *)

%Y Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556-A330561.

%K nonn,easy

%O 2,1

%A _Joseph L. Pe_, Mar 06 2005