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