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A214228 a(n) = gcd(r,2*n+1) where r is 1 + (A143608(i-1) mod (2*n+1)) and A143608(i) is the first zero mod 2*n+1 other than i=0.

%I A214228 #29 Aug 02 2019 03:43:40
%S A214228 1,1,1,1,1,1,5,1,1,7,1,1,1,1,1,1,7,1,13,1,1,5,1,1,3,1,5,1,1,1,7,1,1,
%T A214228 23,1,1,25,7,1,1,1,5,29,1,7,31,5,1,1,1,1,35,1,1,37,1,23,13,7,1,41,1,1,
%U A214228 1,1,7,5,1,1,47,13,1,49,1,1,9,31,1,53
%N A214228 a(n) = gcd(r,2*n+1) where r is 1 + (A143608(i-1) mod (2*n+1)) and A143608(i) is the first zero mod 2*n+1 other than i=0.
%C A214228 It appears that a(n) * gcd(s,2*n+1) is either 2*n+1 or 1; where s is 1 + (A143608(i+1) mod (2*n+1)) and A143608(i) is as stated in the definition.
%e A214228 a(7) = 5 which is a factor of 2*7+1.
%p A214228 A214228 := proc(n)
%p A214228     local i,r ;
%p A214228     i := 1;
%p A214228     while A143608(i) mod (2*n+1) <> 0 do
%p A214228         i := i+1 ;
%p A214228     end do;
%p A214228     r := 1+(A143608(i-1) mod (2*n+1)) ;
%p A214228     gcd(r,2*n+1) ;
%p A214228 end proc: # _R. J. Mathar_, Jul 22 2012
%t A214228 gcdN1[x_,y_] = GCD[x + 1,y]; r0 = 3; Reap[While[r0 < 200, s1=1; s0=0; count=1; While[True, count++; temp=Mod[4*s1 - s0, r0]; If[temp==0, Break[]]; count++; s0 = s1; s1 = temp; temp=Mod[2*s1-s0,r0]; If[temp == 0, Break[]]; s0 = s1; s1 = temp;]; Sow[gcdN1[s1, r0], c]; r0+=2;]][[2,1]]
%Y A214228 Cf. A143608.
%K A214228 nonn
%O A214228 1,7
%A A214228 _Kenneth J Ramsey_, Jul 07 2012