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%I #3 Mar 30 2012 17:22:26
%S 1,1,1,3,1,1,1,3,4,1,5,1,1,3,1,9,1,7,4,3,1,5,1,23,9,3,10,13,13,29,7,
%T 19,5,21,2,17,2,3,7,7,5,5,6,7,1,43,3,59,27,17,4,5,9,7,1,9,2,9,7,29,8,
%U 9,4,25,3,119,2,27,4,29,4,37,5,3,2,5,9,7,10,49,1,35,12,11,6,1,22,1,13,11,4
%N Fractions a(n)/n are such that gcd(a(n),n)=1, a(n) > 0 and a(n) is as small as possible so that the partial sums of the fractions have prime numerator. Let a(1)=1.
%C By Dirichlet's Theorem, it is always possible to find the next term. See A076477 for the list of primes appearing in the numerator. The denominators of these sums are the same as for harmonic numbers, A002805. The sum of the fractions diverges. Is there an upper bound for a(n)/n?
%e a(4) = 3 because 1/4 yields 1/1 + 1/2 + 1/3 + 1/4 = 25/12, but 3/4 yields 1/1 + 1/2 + 1/3 + 3/4 = 31/12.
%t nMax = 100; lst = {1}; numer = {1}; s = 1; Do[k = 1; While[GCD[k, n] > 1 || ! PrimeQ[Numerator[s + k/n]], k++ ]; s = s + k/n; AppendTo[lst, k]; AppendTo[numer, Numerator[s]]; k++, {n, 2, nMax}]; lst
%Y Cf. A076477.
%K nonn,frac
%O 1,4
%A _T. D. Noe_, Oct 14 2002
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