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A165781 a(n) = (2^A002326(n)-1)/(2*n+1).

%I A165781 #14 Jun 04 2020 08:16:41
%S A165781 1,1,3,1,7,93,315,1,15,13797,3,89,41943,9709,9256395,1,31,117,
%T A165781 1857283155,105,25575,381,91,178481,42799,5,84973577874915,19065,4599,
%U A165781 4885260612740877,18900352534538475,1,63,1101298153654301589
%N A165781 a(n) = (2^A002326(n)-1)/(2*n+1).
%C A165781 a(n) = 1  <=>  n is in A000225  <=>  n = 2^k - 1 with k = 0, 1, 2, ... - _M. F. Hasler_, Sep 20 2017
%H A165781 Robert Israel, <a href="/A165781/b165781.txt">Table of n, a(n) for n = 0..1672</a>
%p A165781 A002326 := proc(n) if n = 0 then 1 ; else numtheory[order](2,2*n+1) ; end if ; end proc:
%p A165781 A165781 := proc(n) (2^A002326(n)-1)/(2*n+1) ; end proc:
%p A165781 seq(A165781(n),n=0..60) ; # _R. J. Mathar_, Nov 16 2009
%t A165781 a[n_] := (2^MultiplicativeOrder[2, 2n+1]-1)/(2n+1);
%t A165781 a /@ Range[0, 40] (* _Jean-François Alcover_, Jun 04 2020 *)
%o A165781 (PARI) a(n)=(2^znorder(Mod(2,n=2*n+1))-1)/n \\ _M. F. Hasler_, Sep 20 2017
%Y A165781 Cf. A002326, A053446, A000225.
%K A165781 easy,nonn
%O A165781 0,3
%A A165781 _Ctibor O. Zizka_, Sep 26 2009
%E A165781 Sign in definition and offset corrected by _R. J. Mathar_, Nov 16 2009