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A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1. 33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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 sequences related to Stern's sequences

Index entries for "core" sequences

FORMULA

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

From Henry Bottomley, Dec 18 2000: (Start)

a(n) = A020652(n-1) + A020653(n-1).

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

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 xrange(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

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Last modified December 13 09:55 EST 2017. Contains 295957 sequences.