A125740 Numbers k such that A117731(k) differs from A082687(k).

14, 52, 98, 105, 111, 114, 119, 164, 310, 444, 518, 602, 676, 686, 715, 735, 749, 833, 1220, 1278, 1339, 1474, 1752, 1946, 2023, 2054, 2166, 3016, 3104, 3502, 3568, 3924, 4107, 4308, 4802, 5145, 5243, 5334, 5718, 5831, 6394, 6724, 7550, 8135, 8164, 8767

Offset

1,1

Comments

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1034 (terms below 10^7)

Eric Weisstein's World of Mathematics, Harmonic Number.

Eric Weisstein's World of Mathematics, Hilbert Matrix.

Example

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.

Mathematica

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} ]

Crossrefs

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.

Cf. 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: A269312 A129025 A113907 * A332594 A338165 A217077

Adjacent sequences: A125737 A125738 A125739 * A125741 A125742 A125743

Keyword

nonn,changed

Author

Alexander Adamchuk, Dec 04 2006, Mar 12 2007

Status

approved