login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A125740 Numbers n such that A117731(n) differs from A082687(n). 3
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..46.

Eric Weisstein, The World of Mathematics: Hilbert Matrix.

Eric Weisstein, The World of Mathematics: Harmonic Number

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. 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 A217077 A214659

Adjacent sequences:  A125737 A125738 A125739 * A125741 A125742 A125743

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Dec 04 2006, Mar 12 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 20 06:51 EDT 2020. Contains 337264 sequences. (Running on oeis4.)