A292941 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].

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, 36, 596, 16, 1193, 16, 36, 148, 16, 16, 2387, 298, 72, 32, 4774, 32, 9549, 72, 16, 596, 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

Offset: 1

Author

Antti Karttunen, Sep 28 2017

Keywords

• nonn

Comments

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).

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, A292942, A292943, A292945.

Cf. also A292253, A292263, A292381, A292385.

Programs

Scheme

(define (A292941 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 6)) 1 0) (* 2 (A292941 (A252463 n))))))

Formula

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.

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

a(n) = A292263(n) AND (A292253(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).

a(n) = A292263(n) AND (A292255(n) XOR A292385(n)). [See comments.]

For n >= 0, a(A163511(n)) = A292942(n).

For n >= 1, a(n) + A292943(n) + A292945(n) = A243071(n).