OFFSET
1,2
COMMENTS
See A293248 for the corresponding denominators.
The sequence S is a "rational" variant of A232559.
If r appears in S, then 1/r appears in S.
S is a permutation of the positive rational numbers:
- let f be the function u/v -> (u+1)/v
and g be the function u/v -> u/(v+1),
- let h^k be the k-th iterate of h,
- let r = u/v be a rational number in reduced form,
- without loss of generality, we can assume that u > v,
- according to Dirichlet's theorem on arithmetic progressions, we can choose a prime number p = k*u - 1 > u (where k > 2),
- we also have k*u - 1 > k*v,
- f^(p-1)(1) = p,
- g^(k*v-1)(f^(p-1)(1)) = p / (k*v) (and gcd(p, k*v)=1),
- f(g^(k*v-1)(f^(p-1)(1))) = (p+1) / (k*v) = (k*u) / (k*v) = u/v = r, QED.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A293247
EXAMPLE
S(1) = 1 by definition; so a(1) = 1.
(1+1)/1 = 2 has not yet occurred; so S(2) = 2 and a(2) = 2.
1/(1+1) = 1/2 has not yet occurred; so S(3) = 1/2 and a(3) = 1.
(2+1)/1 = 3 has not yet occurred; so S(4) = 3 and a(4) = 3.
2/(1+1) = 1 has already occurred.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Rémy Sigrist, Oct 03 2017
STATUS
approved