A206297 Position of n in the canonical bijection from the positive integers to the positive rational numbers.

1, 3, 5, 9, 13, 21, 25, 37, 45, 57, 65, 85, 93, 117, 129, 145, 161, 193, 205, 241, 257, 281, 301, 345, 361, 401, 425, 461, 485, 541, 557, 617, 649, 689, 721, 769, 793, 865, 901, 949, 981, 1061, 1085, 1169, 1209, 1257, 1301, 1393, 1425, 1509, 1549

Offset

1,2

Comments

The canonical bijection from the positive integers to the positive rational numbers is given by A038568(n)/A038569(n).

Appears to be a variant of A049691. - R. J. Mathar, Feb 11 2012

It appears that a(n) = 2*A005728(n) - 1. - Chris Boyd, Mar 21 2015

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Example

The canonical bijection starts with 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, so that this sequence starts with 1,3,5,9,13 and A206350 starts with 1,2,4,8,12.

Mathematica

a[n_] := Module[{s = 1, k = 2, j = 1},
  While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
  s = s - 2*EulerPhi[k - 1];
  While[s <= n, If[GCD[j, k - 1] =
    = 1, s = s + 2]; j = j + 1];
  If[s > n + 1, j - 1, k - 1]];
t = Table[a[n], {n, 0, 3000}];   (* A038568 *)
ReplacePart[1 + Flatten[Position[t, 1]], 1, 1]
(* A206297 *)

Prog

(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A206297(n):
    if n == 1:
        return 1
    c, j = 1, 2
    k1 = (n-1)//j
    while k1 > 1:
        j2 = (n-1)//k1 + 1
        c += (j2-j)*(A206297(k1+1)-2)
        j, k1 = j2, (n-1)//j2
    return (n-2)*(n-1)-c+j+2 # Chai Wah Wu, Aug 04 2024

Crossrefs

Cf. A038568, A038569, A206350.

A049691 is an essentially identical sequence. See also A018805.

Sequence in context: A076274 A058989 A049691 * A320596 A227565 A136252

Adjacent sequences: A206294 A206295 A206296 * A206298 A206299 A206300

Keyword

nonn

Author

Clark Kimberling, Feb 06 2012

Status

approved