%I #4 Mar 31 2012 13:20:34

%S 7,13,7,7,37,19,119,41,31,37,37,43,13,7,13,49,7,7,61,71,103,67,73,139,

%T 17,79,19,29,97,103,223,109,37,359,7,49,7,127,953,7,139,41,151,1627,

%U 157,797,179,13,163,13,13,13,13,13,31,31,181,193,199,919,193,211,757,37

%N The ratio of A117731(n) and A082687(n) when they are different.

%C Corresponding numbers n such that A117731(n) differs from A082687(n) are listed in A125740(n) = {14, 52, 98, 105, 111, 114, 119, 164, 310, 444, 518, 602, 676, 686, 715, 735, 749, 833, ...}. a(n) divides A125740(n). Most a(n) are primes.

%C The first composite term in a(n) is a(7) = 119 = 7*17. a(n) is composite for n = {7, 16, 36}. a(16) = a(36) = 49 = 7^2.

%F a(n) = A117731[ A125740(n) ] / A082687[ A125740(n) ].

%e A082687(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, ...}.

%e Thus a(1) = 7 because A117731(n)/A082687(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1,...}.

%t h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f,g], Print[ {n,f/g} ] ], {n,1,10000} ]

%Y Cf. A125740 = numbers n such that A117731(n) differs from A082687(n). Cf. A117731 = Numerator of n*Sum[ 1/(n+k), {k, 1, n} ]. Cf. A082687 = Numerator of Sum[ 1/(n+k), {k, 1, n} ].

%K hard,nonn

%O 1,1

%A _Alexander Adamchuk_, Dec 04 2006