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A111879 Numerators of array which counts positive rational numbers (not including natural numbers).

%I A111879 #16 Jan 21 2025 03:26:27
%S A111879 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 A111879 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 A111879 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 A111879 Numerators of array which counts positive rational numbers (not including natural numbers).
%C A111879 Denominators are given by A111880.
%C A111879 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 A111879 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 A111879 P. Dienes, The Taylor Series, Dover 1957, p. 8, eq.(1).
%H A111879 Wolfdieter Lang, <a href="/A111879/a111879.txt">Array of ratios and more.</a>
%F A111879 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 A111879 Triangle begins:
%e A111879   [1],
%e A111879   [1],
%e A111879   [1, 2, 3],
%e A111879   [1],
%e A111879   [1, 2, 3, 4, 5],
%e A111879   [1, 3, 5],
%e A111879   [1, 2, 4, 5, 7],
%e A111879   [1, 3, 7],
%e A111879   ...
%e A111879 The corresponding ratios are:
%e A111879   [1/2],
%e A111879   [1/3],
%e A111879   [1/4, 2/3, 3/2],
%e A111879   [1/5],
%e A111879   [1/6, 2/5, 3/4, 4/3, 5/2],
%e A111879   [1/7, 3/5, 5/3],
%e A111879   [1/8, 2/7, 4/5, 5/4, 7/2],
%e A111879   [1/9, 3/7, 7/3],
%e A111879   ...
%Y A111879 Row sums give A111881(n)/A069220(n), n>=3, see the W. Lang link.
%Y A111879 Cf. A020652/A020653 if natural numbers are included.
%Y A111879 Cf. A111880.
%K A111879 nonn,easy,frac,tabf
%O A111879 3,4
%A A111879 _Wolfdieter Lang_, Aug 23 2005