A292945 - OEIS

%I #8 Sep 30 2017 16:07:37

%S 0,0,0,0,1,0,2,0,0,2,5,0,10,4,0,0,21,0,42,4,4,10,85,0,0,20,0,8,171,0,

%T 342,0,8,42,1,0,684,84,20,8,1369,8,2738,20,0,170,5477,0,0,0,40,40,

%U 10955,0,8,16,84,342,21911,0,43822,684,8,0,17,16,87644,84,168,2,175289,0,350578,1368,0,168,3,40,701156,16,0,2738,1402313,16,40,5476,340,40

%N Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

%C The AND - XOR formulas are just a restatement of the fact that J(-3|n) = J(-1|n)*J(3|n), i.e., that Jacobi-symbol is multiplicative (also) with respect to its upper argument.

%H Antti Karttunen, <a href="/A292945/b292945.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, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 5 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+5, and 0 otherwise.

%F Also, for n > 1, 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 (A292255(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).

%F a(n) = A292263(n) AND (A292253(n) XOR A292385(n)). [See comments.]

%F For n >= 0, a(A163511(n)) = A292946(n).

%F For n >= 1, A292941(n) + A292943(n) + a(n) = A243071(n).

%o (Scheme) (define (A292945 n) (if (<= n 1) 0 (+ (if (= 5 (modulo n 6)) 1 0) (* 2 (A292945 (A252463 n))))))

%Y Cf. A005940, A163511, A243071, A292941, A292943, A292946.

%Y Cf. also A292253, A292255.

%K nonn

%O 1,7

%A _Antti Karttunen_, Sep 28 2017