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For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the denominator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351705 for the numerators.
4

%I #16 Feb 23 2022 10:51:03

%S 1,1,2,2,1,1,2,2,4,4,4,4,4,4,4,4,1,1,2,2,1,1,2,2,4,4,4,4,4,4,4,4,8,8,

%T 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,1,1,2,2,

%U 1,1,2,2,4,4,4,4,4,4,4,4,1,1,2,2,1,1,2

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

%C 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 Rémy Sigrist, <a href="/A351706/b351706.txt">Table of n, a(n) for n = 0..8191</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dyadic_rational">Dyadic rational</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(A000695(n)) = 1.

%F a(2^k) = A072345(k) for any k >= 0.

%F a(2^k-1) = A016116(k) for any k >= 0.

%e For n = 13:

%e - 13 = 2^0 + 2^2 + 2^3,

%e - A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,

%e - d(13) = 2^0 + 2^1 + 2^-2 = 13/4,

%e - so a(13) = 4.

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

%Y Cf. A000695, A016116, A072345, A351705, A351785, A351786.

%K nonn,base,frac

%O 0,3

%A _Rémy Sigrist_, Feb 16 2022