cp's OEIS Frontend

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.

%I A351705 #15 Feb 23 2022 10:49:07
%S A351705 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,
%T A351705 29,27,31,1,9,5,13,17,25,21,29,3,11,7,15,19,27,23,31,33,41,37,45,49,
%U A351705 57,53,61,35,43,39,47,51,59,55,63,8,9,17,19,10,11,21
%N 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.
%C A351705 The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.
%H A351705 Rémy Sigrist, <a href="/A351705/b351705.txt">Table of n, a(n) for n = 0..8191</a>
%H A351705 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dyadic_rational">Dyadic rational</a>
%H A351705 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A351705 a(A000695(n)) = n.
%F A351705 a(2^k) = A072345(k-1) for any k > 0.
%F A351705 a(2^k-1) = 2^k-1 for any k >= 0.
%F A351705 A000120(a(n)) = A000120(n).
%e A351705 For n = 13:
%e A351705 - 13 = 2^0 + 2^2 + 2^3,
%e A351705 - A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,
%e A351705 - d(13) = 2^0 + 2^1 + 2^-2 = 13/4,
%e A351705 - so a(13) = 13.
%o A351705 (PARI) a(n) = { my (d=0, k); while (n, n-=2^k=valuation(n,2); d+=2^((-1)^k*(k+1)\2)); numerator(d) }
%Y A351705 Cf. A000120, A000695, A072345, A351706, A351785, A351786.
%K A351705 nonn,base,frac
%O A351705 0,4
%A A351705 _Rémy Sigrist_, Feb 16 2022