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A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol.

%I A292253 #13 Sep 30 2017 16:06:21
%S A292253 0,1,2,2,4,4,8,4,4,8,17,8,35,16,8,8,70,8,140,16,16,34,281,16,9,70,8,
%T A292253 32,562,16,1124,16,32,140,17,16,2249,280,68,32,4498,32,8996,68,16,562,
%U A292253 17993,32,19,18,140,140,35986,16,32,64,280,1124,71973,32,143947,2248,32,32,64,64,287894,280,560,34,575789,32,1151579,4498,16,560,34,136,2303158,64
%N A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol.
%C A292253 Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+1 or of the form 12k+11 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right.
%C A292253 The AND - XOR formula(s) just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.
%H A292253 Antti Karttunen, <a href="/A292253/b292253.txt">Table of n, a(n) for n = 1..2048</a>
%H A292253 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A292253 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = +1, and 0 otherwise.
%F A292253 a(n) = A292263(n) AND (A292383(n) XOR A292941(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987). [See comments.]
%F A292253 For n >= 0, a(A163511(n)) = A292254(n).
%F A292253 For n >= 1, a(n) + A292255(n) + A292943(n) = A243071(n).
%o A292253 (Scheme) (define (A292253 n) (if (<= n 2) (- n 1) (+ (if (and (odd? n) (= 1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292253 (A252463 n))))))
%Y A292253 Cf. A005940, A163511, A243071, A292254, A292255, A292263, A292941, A292943, A292383.
%K A292253 nonn
%O A292253 1,3
%A A292253 _Antti Karttunen_, Sep 28 2017