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%I #14 Sep 30 2017 16:07:06
%S 0,1,2,2,4,4,9,4,4,8,18,8,37,18,8,8,74,8,149,16,16,36,298,16,9,74,8,
%T 36,596,16,1193,16,36,148,16,16,2387,298,72,32,4774,32,9549,72,16,596,
%U 19098,32,19,18,148,148,38196,16,33,72,296,1192,76392,32,152785,2386,32,32,72,72,305571,296,596,32,611142,32,1222285,4774,16,596,32
%N a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].
%C Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+1 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 (less significant end).
%C The AND - XOR formulas 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 Antti Karttunen, <a href="/A292941/b292941.txt">Table of n, a(n) for n = 1..2048</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+1, and 0 otherwise.
%F Also, for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = 1], where J is the Jacobi-symbol.
%F a(n) = A292263(n) AND (A292253(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
%F a(n) = A292263(n) AND (A292255(n) XOR A292385(n)). [See comments.]
%F For n >= 0, a(A163511(n)) = A292942(n).
%F For n >= 1, a(n) + A292943(n) + A292945(n) = A243071(n).
%o (Scheme) (define (A292941 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 6)) 1 0) (* 2 (A292941 (A252463 n))))))
%Y Cf. A005940, A163511, A243071, A292942, A292943, A292945.
%Y Cf. also A292253, A292263, A292381, A292385.
%K nonn
%O 1,3
%A _Antti Karttunen_, Sep 28 2017
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