A111879 - OEIS

%I #10 Aug 29 2019 13:10:30

%S 1,1,1,2,3,1,1,2,3,4,5,1,3,5,1,2,4,5,7,1,3,7,1,2,3,4,5,6,7,8,9,1,5,7,

%T 1,2,3,4,5,6,7,8,9,10,11,1,3,5,9,11,1,2,4,7,8,11,13,1,3,5,7,9,11,13,1,

%U 2,3,4,5,6,7,8,9,10,11,12,13,14,15,1,5,7,11,13,1,2,3,4,5,6,7,8,9,10,11,12

%N Numerators of array which counts positive rational numbers (not including natural numbers).

%C Denominators are given by A111880.

%C The sequence of row lengths is [1, 1, 3, 1, 5, 3, 5, 3, 9, 3, 11, 5, 7, 7, ...] = A000010(n)-1 = phi(n)-1, with Euler's totient function phi(n).

%C For n>=3 delete from the list [seq(j/n-j,j=1..n-2)] the natural numbers and the ratios p/q with (p,q) not 1 (p and q not relatively prime, i.e., p and q have a common divisor >1).

%D P. Dienes, The Taylor Series, Dover 1957, p. 8, eq.(1).

%H W. Lang, <a href="/A111879/a111879.txt">Array of ratios and more.</a>

%F a(n, k)=numerator(r(n, k)), n>=3, k=1..phi(n)-1, with phi(n):=A000010(n) (Euler's totient function) and the ratios r(n, k) defined for row n above.

%e [1], [1], [1, 2, 3], [1], [1, 2, 3, 4, 5], [1, 3, 5], [1, 2, 4, 5,

%e 7], [1, 3, 7],...

%e The corresponding ratios are: [1/2], [1/3], [1/4, 2/3, 3/2], [1/5],

%e [1/6, 2/5, 3/4, 4/3, 5/2], [1/7, 3/5, 5/3], [1/8, 2/7, 4/5, 5/4, 7/2], [1/9,

%e 3/7, 7/3],...

%Y Row sums give A111881(n)/A069220(n), n>=3, see the W. Lang link.

%Y Cf. A020652/A020653 if natural numbers are included.

%K nonn,easy,frac,tabf

%O 3,4

%A _Wolfdieter Lang_, Aug 23 2005