1,1

All listed terms are composite.

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, ...}.

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.

Eric Weisstein, The World of Mathematics: Hilbert Matrix.

Eric Weisstein, The World of Mathematics: Harmonic Number

A117731(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 54260455193, ...}.

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

Thus a(1) = 14 because for n<14 A117731(n) = A082687(n) but A117731(14) = 54260455193 differs from A082687(14) = 7751493599.

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} ]

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.

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).

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.

Sequence in context: A007588 A129025 A113907 this_sequence A118856 A118530 A048971

Adjacent sequences: A125737 A125738 A125739 this_sequence A125741 A125742 A125743

nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 04 2006, Mar 12 2007