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%I #6 Jan 01 2020 23:57:23
%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 (Prime(n + 1) - Prime(n))/2 (mod 2).
%C Questions: Is s(n) = (1/n) Sum[a(i),{i,2,n+1)] > 1/2 for all n? Does s(n) --> 1/2 ?
%e (Prime(2 + 1) - Prime(2)) (mod 2) = (5 - 3)/2 (mod 2) = 1 mod 2 = 1. So a(1) = 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
%O 2,1
%A _Joseph L. Pe_, Mar 06 2005
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