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A038566
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Numerators in canonical bijection from positive integers to positive rationals <= 1: arrange fractions by increasing denominator then by increasing numerator:
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25
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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; internal format)
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OFFSET
| 1,4
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COMMENTS
| Also numerators in canonical bijection from positive integers to all positive rational numbers: arrange fractions in triangle in which n-th row the phi(n) contains fractions i/j with GCD(i,j) = 1, i+j=n, i=1,...,n-1, j=n-1,...,1. 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)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 02 2009]
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REFERENCES
| Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
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LINKS
| David Wasserman, Table of n, a(n) for n = 1..100001
W. Lang, Rows of rationals, n=1..24.
Index entries for sequences related to Stern's sequences
Index entries for "core" sequences
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FORMULA
| The n-th "clump" consists of the phi(n) integers <= n and prime to n.
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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:
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;
1,5,7,11,13,17;
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18;
1,3,7,9,11,13,17,19;
...
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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);
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MATHEMATICA
| Flatten[Table[Flatten[Position[GCD[Table[Mod[j, w], {j, 1, w-1}], w], 1]], {w, 1, 100}], 2]
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CROSSREFS
| Cf. A020652, A020653, A038566-A038569, A000010, A002088, A060837, A071970.
A054424 gives mapping to Stern-Brocot tree.
Row sums give rationals A111992(n)/A069220(n), n>=1.
Sequence in context: A132662 A132589 A054843 * A020652 A096107 A128487
Adjacent sequences: A038563 A038564 A038565 * A038567 A038568 A038569
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KEYWORD
| nonn,frac,core,nice,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Erich Friedman (erich.friedman(AT)stetson.edu).
Offset corrected by Max Alekseyev (maxale(AT)gmail.com), Apr 26 2010
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