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A261414 2^n+1 appears in A109732 at position a(n).

%I A261414 #26 Nov 25 2020 05:26:47
%S A261414 2,5,30,38,201,242,689,1806,7175,10839,21474,64607,290563,290579,
%T A261414 581260,872576,2617577,5238258,7858320,19886365,47140605,70713773,
%U A261414 212133736
%N A261414 2^n+1 appears in A109732 at position a(n).
%C A261414 This assumes van der Poorten's conjecture that every odd number does appear in A109732.
%e A261414 A109732(38) = 17 = 2^4+1, so a(4)=38.
%p A261414 m:= 22000: # m is the search limit
%p A261414 b:= proc() true end:
%p A261414 s:= heap[new]((x, y)-> is(x>y), 1):
%p A261414 for n to m do t:= heap[extract](s); b(t):= false;
%p A261414   if t>1 and t-1=2^ilog2(t-1) then print(ilog2(t-1), t, n) fi;
%p A261414   k:= 2*t+1; if b(k) then heap[insert](k, s) fi;
%p A261414   if irem(t, 3, 'k')=0 and b(k) then heap[insert](k, s) fi
%p A261414 od:  # _Alois P. Heinz_, Aug 27 2015
%t A261414 maxVal = 5*10^5; (* 5*10^5 gives 12 terms *)
%t A261414 f[n_] := Module[{lst = {}, x = n}, While[x = 2x+1; x < maxVal, AppendTo[lst, x]]; lst];
%t A261414 M = {1}; pending = f[1];
%t A261414 A261414 = Reap[Print[2]; Sow[2]; While[Length[pending] > 0, next = First[pending]; pending = Rest[pending]; If[!MemberQ[M, next], AppendTo[M, next]; While[Mod[next, 3]==0 && !MemberQ[M, next/3], next = next/3; If[IntegerQ[Log[2, next-1]], Print[an = Length[M]+1]; Sow[an]]; AppendTo[M, next]; pending = Union[pending, f[next]]]]]][[2, 1]] (* _Jean-François Alcover_, Nov 25 2020, after _T. D. Noe_ in A109732 *)
%Y A261414 Cf. A109732, A109734, A109739, A109740, A261412, A261413.
%K A261414 nonn,more
%O A261414 1,1
%A A261414 _N. J. A. Sloane_, Aug 25 2015
%E A261414 a(10)-a(17) from _Alois P. Heinz_, Aug 27 2015
%E A261414 a(18)-a(23) from _Alois P. Heinz_, Aug 28 2015