%I #5 Mar 31 2012 13:20:34
%S 14,52,98,105,111,114,119,164,310,444,518,602,676,686,715,735,749,833,
%T 1220,1278,1339,1474,1752,1946,2023,2054,2166,3016,3104,3502,3568,
%U 3924,4107,4308,4802,5145,5243,5334,5718,5831,6394,6724,7550,8135,8164,8767
%N Numbers n such that A117731(n) differs from A082687(n).
%C All listed terms are composite.
%C The ratio of A117731(n) and A082687(n) when they are different is listed in A125741(n) = A117731[ a(n) ] / A082687[ a(n) ] = {7, 13, 7, 7, 37, 19, 119, 41, 31, 37, 37, 43, 13, 7, 13, 49, 7, 7, 61, 71, 103, 67, 73, 139, ...}.
%C It appears that all (or almost all) members of geometric progressions 2*7^k, 4*13^k, 15*7^k, 3^37^k, 6*19^k, 17*7^k, 4*41^k, 10*31^k, 12*37^k, 55*13^k, 107*7^k, etc. for k>0 are in the sequence.
%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%e A117731(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 54260455193, ...}.
%e A082687(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, ...}.
%e Thus a(1) = 14 because for n<14 A117731(n) = A082687(n) but A117731(14) = 54260455193 differs from A082687(14) = 7751493599.
%t h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f,g], Print[n] ], {n,1,17381} ]
%Y Cf. A117731 = Numerator of n*Sum[ 1/(n+k), {k, 1, n} ]. Cf. A082687 = Numerator of Sum[ 1/(n+k), {k, 1, n} ]. Cf. A125741 = The ratio of A117731(n) and A082687(n) when they are different.
%Y Cf. A082687(n) = numerator of the 2n-th alternating harmonic number H'(2n) = Sum ((-1)^(k+1)/k, k=1..2n). H'(2n) = H(2n) - H(n), where H(n) = Sum (1/k, k=1..n) is the n-th harmonic number. A117731(n) = numerator of the sum of all matrix elements of n X n Hilbert matrix M(i, j) = 1/(i+j-1), (i, j=1..n).
%Y Cf. A126196, A126197, A125581 = numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number.
%K nonn
%O 1,1
%A _Alexander Adamchuk_, Dec 04 2006, Mar 12 2007