A206350 Position of 1/n in the canonical bijection from the positive integers to the positive rational numbers.

1, 2, 4, 8, 12, 20, 24, 36, 44, 56, 64, 84, 92, 116, 128, 144, 160, 192, 204, 240, 256, 280, 300, 344, 360, 400, 424, 460, 484, 540, 556, 616, 648, 688, 720, 768, 792, 864, 900, 948, 980, 1060, 1084, 1168, 1208, 1256, 1300, 1392, 1424, 1508, 1548

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 A049696. - R. J. Mathar, Feb 11 2012

Apparently numbers m such that A071912(m) = 1. - Bill McEachen, Aug 01 2023

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

a(1) = 1, a(n+1) = Sum_{k=1..n} mu(k) * floor(n/k) * floor(1 + n/k), where mu(k) is the Moebius function A008683. - Daniel Suteu, May 28 2018

a(n) = 2*Sum_{k=1..n-1} A000010(k), a(1) = 1. - Robert Israel, Apr 24 2015

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 A206297 starts with 1,3,5,9,13 and this sequence starts with 1,2,4,8,12.

Maple

1, op(2*ListTools:-PartialSums(map(numtheory:-phi, [$1..100]))); # Robert Israel, Apr 24 2015

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[Flatten[Position[t, 1]], 1, 1] (* A206350 *)
(* Second program *)
a[n_]:= If[n==1, 1, 2*Sum[EulerPhi[k], {k, n-1}]];;
Table[a[n], {n, 80}] (* G. C. Greubel, Mar 29 2023 *)

Prog

(Magma) [1] cat [2*(&+[EulerPhi(k): k in [1..n-1]]): n in [2..80]]; // G. C. Greubel, Mar 29 2023
(SageMath)
def A206350(n): return 1 if (n==1) else 2*sum(euler_phi(k) for k in range(1, n))
[A206350(n) for n in range(1, 80)] # G. C. Greubel, Mar 29 2023

Crossrefs

Cf. A000010, A008683, A038568, A038569, A049696, A171503, A206296.

Cf. A071912.

Sequence in context: A110571 A037168 A049696 * A127405 A344932 A136034

Adjacent sequences: A206347 A206348 A206349 * A206351 A206352 A206353

Keyword

nonn

Author

Clark Kimberling, Feb 06 2012

Status

approved