

A020652


Numerators in canonical bijection from positive integers to positive rationals.


28



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

1,3


COMMENTS

a(A002088(n)) = 1 for n > 1.  Reinhard Zumkeller, Jul 29 2012
When read as an irregular table with each 1 entry starting a new row, then the nth row consists of the set of multiplicative units of Z_{n+1}. These rows form a group under multiplication mod n.  Tom Edgar, Aug 20 2013
The pair of sequences A020652/A020653 is defined by ordering the positive fractions p/q (reduced to lowest terms) by increasing p+q, then increasing p: 1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 2/5, 3/4, 4/3, 5/2; etc. For given p+q, there are A000010(p+q) fractions, listed starting at index A002088(p+q1).  M. F. Hasler, Mar 06 2020


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.
Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 7980.
H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.


LINKS

David Wasserman, Table of n, a(n) for n = 1..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


EXAMPLE

Arrange positive fractions < 1 by increasing denominator then by increasing numerator: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6 ... (this is A020652/A038567).  William Rex Marshall, Dec 16 2010


MAPLE

with (numtheory): A020652 := proc (n) local sum, j, k; sum := 0: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: sum := sum  phi(k1): j := 1; while sum < n do: if gcd(j, k1) = 1 then sum := sum + 1: fi: j := j+1: od: RETURN (j1): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 06 2001


MATHEMATICA

Reap[Do[If[GCD[num, den] == 1, Sow[num]], {den, 1, 20}, {num, 1, den1}] ][[2, 1]] (* JeanFrançois Alcover, Oct 22 2012 *)


PROG

(Haskell)
a020652 n = a020652_list !! (n1)
a020652_list = map fst [(u, v)  v < [1..], u < [1..v1], gcd u v == 1]
 Reinhard Zumkeller, Jul 29 2012
(PARI) a(n)=my(s, j=1, k=1); while(s<n, s+=eulerphi(k++); ); s=eulerphi(k); while(s<n, if(gcd(j, k)==1, s++); j++); j1 \\ Charles R Greathouse IV, Feb 07 2013
(Python)
from sympy import totient, gcd
def a(n):
s=0
k=2
while s<n:
s+=totient(k)
k+=1
s=totient(k  1)
j=1
while s<n:
if gcd(j, k  1)==1:
s+=1
j+=1
return j  1
print [a(n) for n in range(1, 101)] # Indranil Ghosh, May 23 2017, after Ulrich Schimke's MAPLE code


CROSSREFS

Essentially the same as A038566, which is the main entry for this sequence.
Cf. A020653, A038567A038569, A182972A182976.
A054424 gives mapping to SternBrocot tree.
Cf. A037161.
Sequence in context: A231631 A280700 A038566 * A293248 A096107 A329585
Adjacent sequences: A020649 A020650 A020651 * A020653 A020654 A020655


KEYWORD

nonn,frac,core,nice


AUTHOR

David W. Wilson


STATUS

approved



