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A038566 Numerators in canonical bijection from positive integers to positive rationals <= 1: arrange fractions by increasing denominator then by increasing numerator: 63
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

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

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

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 sequences related to Stern's sequences

Index entries for "core" sequences

FORMULA

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

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 A038566/A020653).

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

CROSSREFS

Cf. A020652, A020653, A038566, A038567, A038568, A038569, A000010 (row lengths), A002088, A060837, A071970, A002260.

A054424 gives mapping to Stern-Brocot tree.

Row sums give rationals A111992(n)/A069220(n), n>=1.

A112484 (primes, rows n >=3).

Sequence in context: A277427 A231631 A280700 * A020652 A293248 A096107

Adjacent sequences:  A038563 A038564 A038565 * A038567 A038568 A038569

KEYWORD

nonn,frac,core,nice,tabf

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Erich Friedman

Offset corrected by Max Alekseyev, Apr 26 2010

STATUS

approved

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Last modified November 19 01:40 EST 2017. Contains 294912 sequences.