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A214229 a(n) equals gcd(r,2*n+1) where r is 1 + (A143608(i+1) mod (2*n+1)) where A143608(i) is the first zero mod 2n+1 other than 0.

%I A214229 #29 Aug 02 2019 05:27:06
%S A214229 3,5,1,9,11,13,3,17,19,3,1,25,27,29,1,33,5,37,3,1,43,9,1,1,17,53,11,
%T A214229 57,59,61,9,65,67,3,1,73,3,11,1,81,83,17,3,89,13,3,19,97,99,101,1,3,
%U A214229 107,109,3,113,5,9,17,121,3,125,1,129,131,19
%N A214229 a(n) equals gcd(r,2*n+1) where r is 1 + (A143608(i+1) mod (2*n+1)) where A143608(i) is the first zero mod 2n+1 other than 0.
%C A214229 It appears that a(n) * b(n) either equals 2*n+1 or 1 where b is the companion sequence A214228.
%e A214229 a(7) = 3 which is a factor of 2*7 + 1.
%p A214229 A214229 := proc(n)
%p A214229     local i,r ;
%p A214229     i := 1;
%p A214229     while A143608(i) mod (2*n+1) <> 0 do
%p A214229         i := i+1 ;
%p A214229     end do;
%p A214229     r := 1+(A143608(i+1) mod (2*n+1)) ;
%p A214229     gcd(r,2*n+1) ;
%p A214229 end proc: # _R. J. Mathar_, Jul 22 2012
%t A214229 gcdN2[x_,y_] = GCD[y - x + 1,y];
%t A214229 r0 = 3;
%t A214229 table=Reap[While[r0 < 200,s1=1;s0=0;count=1;While[True,count++;temp=Mod[4*s1 - s0,r0];
%t A214229 If[temp==0,Break[]];count++;s0 = s1; s1 = temp;
%t A214229 temp=Mod[2*s1-s0,r0];If[temp == 0,Break[]];s0 = s1;s1 = temp;];
%t A214229 Sow[gcdN2[s1,r0],d];
%t A214229 r0+=2;]][[2]];
%t A214229 table
%K A214229 nonn
%O A214229 1,1
%A A214229 _Kenneth J Ramsey_, Jul 07 2012