A351705 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the numerator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351706 for the denominators.

0, 1, 1, 3, 2, 3, 5, 7, 1, 5, 3, 7, 9, 13, 11, 15, 4, 5, 9, 11, 6, 7, 13, 15, 17, 21, 19, 23, 25, 29, 27, 31, 1, 9, 5, 13, 17, 25, 21, 29, 3, 11, 7, 15, 19, 27, 23, 31, 33, 41, 37, 45, 49, 57, 53, 61, 35, 43, 39, 47, 51, 59, 55, 63, 8, 9, 17, 19, 10, 11, 21

Offset

0,4

Comments

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Wikipedia, Dyadic rational

Index entries for sequences related to binary expansion of n

Formula

a(A000695(n)) = n.

a(2^k) = A072345(k-1) for any k > 0.

a(2^k-1) = 2^k-1 for any k >= 0.

A000120(a(n)) = A000120(n).

Example

For n = 13:
- 13 = 2^0 + 2^2 + 2^3,
- A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,
- d(13) = 2^0 + 2^1 + 2^-2 = 13/4,
- so a(13) = 13.

Prog

(PARI) a(n) = { my (d=0, k); while (n, n-=2^k=valuation(n, 2); d+=2^((-1)^k*(k+1)\2)); numerator(d) }

Crossrefs

Cf. A000120, A000695, A072345, A351706, A351785, A351786.

Sequence in context: A347266 A050060 A247829 * A246592 A241816 A243109

Adjacent sequences: A351702 A351703 A351704 * A351706 A351707 A351708

Keyword

nonn,base,frac

Author

Rémy Sigrist, Feb 16 2022

Status

approved