A038566 - OEIS

A038566

Numerators in canonical bijection from positive integers to positive rationals <= 1: arrange fractions by increasing denominator then by increasing numerator.

1, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

For denominators see A038567.

Row n has length A000010(n).

Also numerators in canonical bijection from positive integers to all positive rational numbers: arrange fractions in triangle in which in the n-th row the phi(n) numbers are the fractions i/j with gcd(i,j) = 1, i+j=n, i=1..n-1, j=n-1..1. n>=2. Denominators (A020653) are obtained by reversing each row.

Also triangle in which n-th row gives phi(n) numbers between 1 and n that are relatively prime to n.

A038610(n) = least common multiple of n-th row. - Reinhard Zumkeller, Sep 21 2013

Row n has sum A023896(n). - Jamie Morken, Dec 17 2019

This irregular triangle gives in row n the smallest positive reduced residue system modulo n, for n >= 1. If one takes 0 for n = 1 it becomes the smallest nonnegative residue system modulo n. - Wolfdieter Lang, Feb 29 2020

REFERENCES

Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.

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

LINKS

David Wasserman, Table of n, a(n) for n = 1..100001

Wolfdieter Lang, Rows of rationals, n=2..25.

Index entries for "core" sequences

Index entries for sequences related to enumerating the rationals

Index entries for sequences related to Stern's sequences

FORMULA

The n-th "clump" consists of the phi(n) integers <= n and prime to n.

a(n) = A002260(A169581(n)). - Reinhard Zumkeller, Dec 02 2009

a(n+1) = A020652(n) for n > 1. - Georg Fischer, Oct 27 2020

EXAMPLE

The beginning of the list of positive rationals <= 1: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, .... This is A038566/A038567.

The beginning of the triangle giving all positive rationals: 1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 1/6, 2/5, 3/4, 4/3, 5/2, 6/1; .... This is A020652/A020653, with A020652(n) = A038566(n+1). [Corrected by M. F. Hasler, Mar 06 2020]

The beginning of the triangle in which n-th row gives numbers between 1 and n that are relatively prime to n:

n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1: 1

2: 1

3: 1 2

4: 1 3

5: 1 2 3 4

6: 1 5

7: 1 2 3 4 5 6

8: 1 3 5 7

9: 1 2 4 5 7 8

10: 1 3 7 9

11: 1 2 3 4 5 6 7 8 9 10

12: 1 5 7 11

13: 1 2 3 4 5 6 7 8 9 10 11 12

14: 1 3 5 9 11 13

15: 1 2 4 7 8 11 13 14

16: 1 3 5 7 9 11 13 15

17: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18: 1 5 7 11 13 17

19: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

20: 1 3 7 9 11 13 17 19

... Reformatted. - Wolfdieter Lang, Jan 18 2017

------------------------------------------------------

MAPLE

s := proc(n) local i, j, k, ans; i := 0; ans := [ ]; for j while i<n do for k to j do if gcd(j, k) = 1 then ans := [ op(ans), k ]; i := i+1 fi od od; RETURN(ans); end; s(100);

MATHEMATICA

Flatten[Table[Flatten[Position[GCD[Table[Mod[j, w], {j, 1, w-1}], w], 1]], {w, 1, 100}], 2]

PROG

(Haskell)

a038566 n k = a038566_tabf !! (n-1) !! (k-1)

a038566_row n = a038566_tabf !! (n-1)

a038566_tabf=

zipWith (\v ws -> filter ((== 1) . (gcd v)) ws) [1..] a002260_tabl

a038566_list = concat a038566_tabf

-- Reinhard Zumkeller, Sep 21 2013, Feb 23 2012

(PARI) first(n)=my(v=List(), i, j); while(i<n, for(k=1, j, if(gcd(j, k)==1, listput(v, k); i++)); j++); Vec(v) \\ Charles R Greathouse IV, Feb 07 2013

(PARI) row(n) = select(x->gcd(n, x)==1, [1..n]); \\ Michel Marcus, May 05 2020

(SageMath)

def aRow(n):

if n == 1: return 1

return [k for k in ZZ(n).coprime_integers(n+1)]

print(flatten([aRow(n) for n in range(1, 18)])) # Peter Luschny, Aug 17 2020

CROSSREFS