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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. 87
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
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
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: A231631 A280700 A356149 * A020652 A293248 A096107
KEYWORD
nonn,frac,core,nice,tabf
AUTHOR
EXTENSIONS
More terms from Erich Friedman
Offset corrected by Max Alekseyev, Apr 26 2010
STATUS
approved

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Last modified July 15 16:36 EDT 2024. Contains 374333 sequences. (Running on oeis4.)