Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[CalkinWilfSequence] instead.

The function ithrational generates the ith positive rational number as illustrated by the first example below. That is, this function generates all positive rational numbers in a non-repeating sequence.

The ordering of the rationals satisfies ithrational(2^(i-1)) = 1/i and ithrational(2^i-1) = i.

The command with(numtheory,ithrational) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

$\mathrm{seq}\left(\mathrm{ithrational}\left(i\right)\,i=0..10\right)$

$0,1,\frac{1}{2},2,\frac{1}{3},\frac{3}{2},\frac{2}{3},3,\frac{1}{4},\frac{4}{3},\frac{3}{5}$

$\mathrm{seq}\left(\mathrm{ithrational}\left(2^i-1\right)\,i=1..10\right)$

$1,2,3,4,5,6,7,8,9,10$

$\mathrm{seq}\left(\mathrm{ithrational}\left(2^{i-1}\right)\,i=1..10\right)$

$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{9},\frac{1}{10}$

$\mathrm{ithrational}\left(1234321\right)$

$\frac{7461}{5875}$