0,4

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.

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

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

David Wasserman, Table of n, a(n) for n = 0..100000

W. Lang, Rows of rationals, n=1..24.

Index entries for sequences related to Stern's sequences

Index entries for "core" sequences

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

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; ...

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);

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

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 this_sequence A020652 A096107 A128487

Adjacent sequences: A038563 A038564 A038565 this_sequence A038567 A038568 A038569

nonn,frac,core,nice,tabf

njas

More terms from Erich Friedman (erich.friedman(AT)stetson.edu).