

A038567


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


39



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
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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 (197677), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (197778), 122123.
H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.


LINKS



FORMULA



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 (k1): 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}] (* JeanFrançois 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..]
(Python)
from sympy import totient
def a(n):
s=1
while sum(totient(i) for i in range(1, s + 1))<n: s+=1


CROSSREFS

A054427 gives mapping to SternBrocot tree.


KEYWORD

nonn,frac,core,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



