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A256992 Position of n in either of the complementary sequences, A005187 or A055938: a(n) = A213714(n) + A234017(n).

%I A256992 #13 Dec 12 2016 09:06:44
%S A256992 1,1,2,3,2,3,4,5,4,6,7,5,6,7,8,9,8,10,11,9,10,12,13,11,14,15,12,13,14,
%T A256992 15,16,17,16,18,19,17,18,20,21,19,22,23,20,21,22,24,25,23,26,27,24,25,
%U A256992 28,29,26,30,31,27,28,29,30,31,32,33,32,34,35,33,34,36,37,35,38,39,36,37,38,40,41,39,42,43,40,41,44,45,42
%N A256992 Position of n in either of the complementary sequences, A005187 or A055938: a(n) = A213714(n) + A234017(n).
%C A256992 In other words, if n = A005187(k) for some k >= 1, then a(n) = k, otherwise it must be that n = A055938(h) for some h, and then a(n) = h.
%C A256992 Each n occurs exactly twice, first at a(A005187(n)), then at a(A055938(n)). Cf. also A257126.
%C A256992 When iterating a(n), a(a(n)), a(a(a(n))), etc, A256993(n) gives the number of steps to reach one, from any starting value n >= 1.
%H A256992 Antti Karttunen, <a href="/A256992/b256992.txt">Table of n, a(n) for n = 1..16384</a>
%F A256992 a(n) = A213714(n) + A234017(n).
%F A256992 a(n) = A256991(n) + A079559(n).
%F A256992 If A079559(n) = 1, a(n) = A213714(n), otherwise a(n) = A234017(n).
%t A256992 With[{nn = 92}, Function[{g, h}, Flatten@ Table[If[MemberQ[g, n], First@ Position[g, n] - 1, First@ Position[h, n]], {n, Min[Length /@ {g, h}]}]] @@ {Table[2 n - DigitCount[2 n, 2, 1], {n, 0, nn}], Complement[Range@ Last@ #, #] &@ Table[IntegerExponent[(2 n)!, 2], {n, 0, nn}]} ] (* _Michael De Vlieger_, Dec 12 2016, after _Harvey P. Dale_ at A005187 and _Jean-François Alcover_ at A055938 *)
%o A256992 (Scheme)
%o A256992 (define (A256992 n) (+ (A213714 n) (A234017 n)))
%o A256992 (define (A256992 n) (if (not (zero? (A079559 n))) (A213714 n) (A234017 n)))
%Y A256992 Cf. A005187, A055938, A079559, A213714, A234017.
%Y A256992 Cf. also A256991 (variant), A256993, A257126.
%K A256992 nonn
%O A256992 1,3
%A A256992 _Antti Karttunen_, Apr 15 2015