A292255 a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == -1], where J is the Jacobi-symbol.

0, 0, 0, 0, 1, 0, 3, 0, 0, 2, 6, 0, 12, 6, 0, 0, 25, 0, 51, 4, 4, 12, 102, 0, 0, 24, 0, 12, 205, 0, 411, 0, 12, 50, 0, 0, 822, 102, 24, 8, 1645, 8, 3291, 24, 0, 204, 6582, 0, 0, 0, 48, 48, 13165, 0, 9, 24, 100, 410, 26330, 0, 52660, 822, 8, 0, 25, 24, 105321, 100, 204, 0, 210642, 0, 421284, 1644, 0, 204, 1, 48, 842569, 16, 0, 3290, 1685138, 16, 48, 6582

Offset: 1

Author

Antti Karttunen, Sep 28 2017

Keywords

• nonn

Comments

Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+5 or of the form 12k+7 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2048
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A005940, A163511, A243071, A292253, A292256, A292263, A292943.

Cf. also A292941, A292945.

Programs

Scheme

(define (A292255 n) (if (<= n 1) 0 (+ (if (and (odd? n) (= -1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292255 (A252463 n))))))

Formula

a(1) = 0, and for n > 1, 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.

a(n) = A292263(n) AND (A292383(n) XOR A292945(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).

a(n) = A292263(n) AND (A292385(n) XOR A292941(n)). [See comments.]

For n >= 0, a(A163511(n)) = A292256(n).

For n >= 1, a(n) + A292253(n) + A292943(n) = A243071(n).