login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

For any nonnegative real number x, let f(x) be the real number obtained by replacing in the binary expansions of the integer and fractional parts of x each finite run of k consecutive equal bits b with a run of k-(-1)^k consecutive bits b; a(n) is the numerator of f(1/n).
2

%I #15 Feb 27 2020 23:21:28

%S 3,3,1,3,1,2,3,3,1,1,13,1,7,3,1,3,1,2,77,1,1,26,203,1,817,14,109,3,

%T 1037,2,3,3,1,1,1297,1,20275,77,155,1,17,1,13,13,67,203,6716227,1,

%U 421735,817,17,7,2306997,109,55739,3,49,1037,818712813,1,138203853,3

%N For any nonnegative real number x, let f(x) be the real number obtained by replacing in the binary expansions of the integer and fractional parts of x each finite run of k consecutive equal bits b with a run of k-(-1)^k consecutive bits b; a(n) is the numerator of f(1/n).

%C When computing f(x), we consider the unique binary representation of x where the fractional part of x does not eventually end with repeating ones.

%C The function f establishes a self-inverse bijection:

%C - over the nonnegative real numbers,

%C - over the nonnegative real numbers in the half-open interval [0,1),

%C - over the nonnegative rational numbers,

%C - over the nonnegative rational numbers in the half-open interval [0,1),

%C - over the nonnegative integers (for any n >= 0, f(n) = A162853(n)).

%C The function f has only one fixed point: f(0) = 0.

%H Rémy Sigrist, <a href="/A323626/a323626.png">Representation of f in the half-open interval [0,1)</a>

%H Rémy Sigrist, <a href="/A323626/a323626_1.gp.txt">PARI program for A323626</a>

%F a(2^k) = 3 for any k >= 0.

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

%e The first terms of the sequence, alongside f(1/n) and the binary representations of 1/n and of f(1/n) with periodic part in parentheses, are:

%e n a(n) f(1/n) bin(1/n) bin(f(1/n))

%e -- ---- ------- ---------------------- ------------------------

%e 1 3 3 1.(0) 11.(0)

%e 2 3 3/4 0.1(0) 0.11(0)

%e 3 1 1/5 0.(01) 0.(0011)

%e 4 3 3/16 0.01(0) 0.0011(0)

%e 5 1 1/3 0.(0011) 0.(01)

%e 6 2 2/5 0.0(01) 0.(0110)

%e 7 3 3/7 0.(001) 0.(011)

%e 8 3 3/8 0.001(0) 0.011(0)

%e 9 1 1/17 0.(000111) 0.(00001111)

%e 10 1 1/24 0.0(0011) 0.000(01)

%e 11 13 13/257 0.(0001011101) 0.(0000110011110011)

%e 12 1 1/20 0.00(01) 0.00(0011)

%e 13 7 7/129 0.(000100111011) 0.(00001101111001)

%e 14 3 3/56 0.0(001) 0.000(011)

%e 15 1 1/21 0.(0001) 0.(000011)

%e 16 3 3/64 0.0001(0) 0.000011(0)

%e 17 1 1/9 0.(00001111) 0.(000111)

%e 18 2 2/17 0.0(000111) 0.(00011110)

%e 19 77 77/1025 0.(000011010111100101) 0.(00010011001110110011)

%e 20 1 1/12 0.00(0011) 0.00(01)

%o (PARI) See Links section.

%Y See A323627 for the corresponding denominators.

%Y Cf. A162853.

%K nonn,frac,base

%O 1,1

%A _Rémy Sigrist_, Jan 20 2019