A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1.

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16

Offset

0,2

Comments

n occurs phi(n) times (cf. A000010).

Least k such that phi(1) + phi(2) + phi(3) + ... + phi(k) >= n. - Benoit Cloitre, Sep 17 2002

Sum of numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed. - Ron R. King, Mar 07 2009 [This applies to a(1, 2, ...) without initial term a(0) = 1 which could correspond to 0/1. - Editor's Note.]

Care has to be taken in considering the offset which may be 0 or 1 in related sequences (see crossrefs), e.g., A038568 & A038569 also have offset 0, in A038566 offset has been changed to 1. - M. F. Hasler, Oct 18 2021

References

S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.

H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.

Links

David Wasserman, Table of n, a(n) for n = 0..100000

Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.

Index entries for "core" sequences

Index entries for sequences related to enumerating the rationals

Index entries for sequences related to Stern's sequences

Formula

From Henry Bottomley, Dec 18 2000: (Start)

a(n) = A020652(n) + A020653(n) for all n > 0, e.g., a(1) = 2 = 1 + 1 = A020652(1) + A020653(1). [Corrected and edited by M. F. Hasler, Dec 10 2021]

n = a(A015614(n)) = a(A002088(n)) - 1 = a(A002088(n-1)). (End)

a(n) = A002024(A169581(n)). - Reinhard Zumkeller, Dec 02 2009

a(A002088(n)) = n for n > 1. - Reinhard Zumkeller, Jul 29 2012

a(n) = A071912(2*n+1). - Reinhard Zumkeller, Dec 16 2013

Example

Arrange fractions by increasing denominator then by increasing numerator: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ...: this is A038566/A038567.

Maple

with (numtheory): A038567 := proc (n) local sum, k; sum := 1: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

Mathematica

a[n_] := (k = 0; While[ Total[ EulerPhi[ Range[k]]] <= n, k++]; k); Table[ a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 08 2011, after Pari *)
Flatten[Table[Table[n, {EulerPhi[n]}], {n, 20}]] (* Harvey P. Dale, Mar 12 2013 *)

Prog

(PARI) a(n)=if(n<0, 0, s=1; while(sum(i=1, s, eulerphi(i))<n, s++); s)
(Haskell)
import Data.List (genericTake)
a038567 n = a038567_list !! n
a038567_list = concatMap (\x -> genericTake (a000010 x) $ repeat x) [1..]
-- Reinhard Zumkeller, Dec 16 2013, Jul 29 2012
(Python)
from sympy import totient
def a(n):
    s=1
    while sum(totient(i) for i in range(1, s + 1))<n: s+=1
    return s # Indranil Ghosh, May 23 2017

Crossrefs

Cf. A020652, A020653, A038566 - A038569, A182972, A182973 - A182976.

A054427 gives mapping to Stern-Brocot tree.

Cf. A037162.

Sequence in context: A126236 A198194 A073047 * A185195 A192512 A036234

Adjacent sequences: A038564 A038565 A038566 * A038568 A038569 A038570

Keyword

nonn,frac,core,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Erich Friedman

Status

approved