A117731 -id:A117731 - cp's OEIS frontend

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

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

Alexander Adamchuk, Dec 04 2006, Mar 12 2007

nonn

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.

			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.

Eric Weisstein, The World of Mathematics: Hilbert Matrix.
Eric Weisstein, The World of Mathematics: Harmonic Number

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.

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} ]

A125741 The ratio of A117731(n) and A082687(n) when they are different.

Original entry on oeis.org

7, 13, 7, 7, 37, 19, 119, 41, 31, 37, 37, 43, 13, 7, 13, 49, 7, 7, 61, 71, 103, 67, 73, 139, 17, 79, 19, 29, 97, 103, 223, 109, 37, 359, 7, 49, 7, 127, 953, 7, 139, 41, 151, 1627, 157, 797, 179, 13, 163, 13, 13, 13, 13, 13, 31, 31, 181, 193, 199, 919, 193, 211, 757, 37

Offset: 1

Alexander Adamchuk, Dec 04 2006

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.

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.

			A082687(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, ...}.
Thus a(1) = 7 because A117731(n)/A082687(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1,...}.

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} ].

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} ]

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

A119030 Difference between numerator and denominator of the sum of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n), A117731[n] - A117664[n].

Original entry on oeis.org

0, 4, 27, 428, 1375, 15797, 211631, 86540, 1496205, 144045379, 145607407, 3378951221, 17021747431, 51392118293, 214084856611, 13337033800292, 13393340889767, 94103945740529, 3493457389196573, 3503912518228613

Offset: 1

Alexander Adamchuk, Jul 22 2006

nonn

p^3 divides a(p^k) for prime p>2 and integer k>0.

Cf. A117731, A117664.

Numerator[Table[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}],{n,1,25}]] - Denominator[Table[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}],{n,1,25}]]

a(n) = Numerator[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]] - Denominator[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]]. a(n) = A117731[n] - A117664[n].

A126563 Numbers k such that the ratio of A117731(k) and A082687(k) is composite.

Original entry on oeis.org

119, 735, 5145, 36015, 252105, 1764735, 12353145

Offset: 1

Alexander Adamchuk, Mar 12 2007, Jun 09 2007

a(1) = 7*17, a(2) = 3*5*7^2, a(3) = 3*5*7^3.
Corresponding composite terms in A125741 are {119, 49, 49, 49, 49, 49, 49, ...}.
A125741(n) is composite for n = {7, 16, 36, 91, 226, 510, 1131, ...}.

Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A082687, A117731, A125581, A125740, A125741, A126196, A126197.

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

f(n) = sum(k=1, n, 1/(n+k));
isok(k) = my(fk = f(k), q = numerator(k*fk)/numerator(fk)); (q!=1) && !isprime(q); \\ Michel Marcus, Mar 08 2023

Edited by Max Alekseyev, Jul 12 2019

a(5)-a(7) from Jinyuan Wang, Jul 10 2025

A082687 Numerator of Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193

Offset: 1

Benoit Cloitre, Apr 12 2003

Numerator of Sum_{k=0..n-1} 1/((k+1)(2k+1)) (denominator is A111876). - Paul Barry, Aug 19 2005
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). - Alexander Adamchuk, Apr 11 2006
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_{k=1..n} 1/k is the n-th Harmonic Number. - Alexander Adamchuk, Apr 11 2006
a(n) almost always equals A117731(n) = numerator(n*Sum_{k=1..n} 1/(n+k)) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)) but differs for n = 14, 53, 98, 105, 111, 114, 119, 164. - Alexander Adamchuk, Jul 16 2006
Sum_{k=1..n} 1/(n+k) = n!^2 *Sum_{j=1..n} (-1)^(j+1) /((n+j)!(n-j)!j). - Leroy Quet, May 20 2007
Seems to be the denominator of the harmonic mean of the first n hexagonal numbers. - Colin Barker, Nov 19 2014
Numerator of 2*n*binomial(2*n,n)*Sum_{k = 0..n-1} (-1)^k* binomial(n-1,k)/(n+k+1)^2. Cf. A049281. - Peter Bala, Feb 21 2017
From Peter Bala, Feb 16 2022: (Start)
2*Sum_{k = 1..n} 1/(n+k) = 1 + 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + - .... Cf. A101028.
2*Sum_{k = 1..n} 1/(n+k) = n - (1 + 1/2^2)*n*(n-1)/(n+1) + (1/2^2 + 1/3^2)*n*(n-1)*(n-2)/((n+1)*(n+2)) - (1/3^2 + 1/4^2)*n*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + (1/4^2 + 1/5^2)*n*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - + .... Cf. A007406 and A120778.
These identities allow us to extend the definition of Sum_{k = 1..n} 1/(n+k) to non-integral values of n. (End)

			H'(2n) = H(2n) - H(n) = {1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, 95549/144144, 1632341/2450448, 155685007/232792560, ...}, where H(n) = A001008/A002805.
n=2: HilbertMatrix(n,n)
   1  1/2
  1/2 1/3
so a(2) = Numerator(1 + 1/2 + 1/2 + 1/3) = Numerator(7/3) = 7.
The n X n Hilbert matrix begins:
   1   1/2  1/3  1/4  1/5  1/6  1/7  1/8  ...
  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  ...
  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
  1/4  1/5  1/6  1/7  1/8  1/9  1/10 1/11 ...
  1/5  1/6  1/7  1/8  1/9  1/10 1/11 1/12 ...
  1/6  1/7  1/8  1/9  1/10 1/11 1/12 1/13 ...

T. D. Noe, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Bisection of A058313, A082688 (denominators).
Cf. A001008, A002805, A005249, A007406, A058312.
Cf. A086881, A098118, A101028, A117731, A120778.

[Numerator((HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023

a := n -> numer(harmonic(2*n) - harmonic(n)):
seq(a(n), n=1..20); # Peter Luschny, Nov 02 2017

Numerator[Sum[1/k,{k,1,2*n}] - Sum[1/k,{k,1,n}]] (* Alexander Adamchuk, Apr 11 2006 *)
Table[Numerator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 20}] (* Alexander Adamchuk, Apr 11 2006 *)
Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)

a(n) = numerator(sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

[numerator(harmonic_number(2*n,1) - harmonic_number(n,1)) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

Limit_{n -> oo} Sum_{k=1..n} 1/(n+k) = log(2).
Numerator of Psi(2*n+1) - Psi(n+1). - Vladeta Jovovic, Aug 24 2003
a(n) = numerator((Sum_{k=1..2*n} 1/k) - Sum_{k=1..n} 1/k). - Alexander Adamchuk, Apr 11 2006
a(n) = numerator(Sum_{j=1..n} (Sum_{i=1..n} 1/(i+j-1))). - Alexander Adamchuk, Apr 11 2006
The o.g.f for Sum_{k=1..n} 1/(n+k) is f(x) = (sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) + log(1-x))/(2*x*(1-x)).

A117664 Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.

Original entry on oeis.org

1, 3, 10, 105, 252, 2310, 25740, 9009, 136136, 11639628, 10581480, 223092870, 1029659400, 2868336900, 11090902680, 644658718275, 606737617200, 4011209802600, 140603459396400, 133573286426580, 5215718803323600

Offset: 1

Alexander Adamchuk, Apr 11 2006

nonn

Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) = A117731(n) / A117664(n) = 2n * H'(2n) = 2n * A058313(2n) / A058312(2n), where H'(2n) is 2n-th alternating sign Harmonic Number. H'(2n) = H(2n) - H(n), where H(n) is n-th Harmonic Number. - Alexander Adamchuk, Apr 23 2006

			For n=2, the 2 X 2 Hilbert matrix is [1, 1/2; 1/2, 1/3], so a(2) = denominator(1 + 1/2 + 1/2 + 1/3) = denominator(7/3) = 3.
The n X n Hilbert matrix begins:
    1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
  ...

Eric Weisstein's World of Mathematics, Hilbert Matrix.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A091342, A098118, A111876, A082687, A086881, A005249, A001008, A002805.

Numerator is A117731(n).

Table[Denominator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 30}]

a(n) = A111876(n-1)/n.
a(n) = denominator( Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) ). Numerator is A117731(n). - Alexander Adamchuk, Apr 23 2006
a(n) = denominator( Sum_{k=1..n} (2*k)/(n+k) ). - Peter Bala, Oct 10 2021

A296519 Denominator of n*Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

2, 6, 20, 210, 504, 4620, 51480, 18018, 272272, 23279256, 21162960, 446185740, 2059318800, 5736673800, 22181805360, 1289317436550, 1213475234400, 8022419605200, 281206918792800, 267146572853160, 10431437606647200, 428163098127382800, 409547311252279200

Offset: 1

Eric W. Weisstein, Dec 14 2017

a(n) is divisible by all primes p such that the numerator of Sum_{n < k*p <= n} 1/k is not divisible by p, in particular by all primes from n+1 to 2*n-1. - Robert Israel, May 21 2020

			The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/a(n).

Robert Israel, Table of n, a(n) for n = 1..1155

Cf. A111876, A117731 (numerators), A117664.

[Denominator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023

N:= 30: # for a(1)..a(N)
H:= ListTools:-PartialSums([seq(1/i,i=1..2*N)]):
map(n -> denom(n*(H[2*n]-H[n])), [$1..N]); # Robert Israel, May 21 2020

Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 30}] // Denominator

a(n) = denominator(n*sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

[denominator(n*(harmonic_number(2*n,1) - harmonic_number(n,1))) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

From G. C. Greubel, Jul 24 2023: (Start)
a(n) = 2*A117664(n).
a(n) = 2*A111876(n-1)/n. (End)

A129764 Numerator of the sum of all elements of n X n X n cubic array M[i,j,k] = 1/(i+j+k-2).

Original entry on oeis.org

1, 15, 1133, 1177, 129149, 349673, 57087959, 345322023, 14272692271, 40165727117, 217549734472087, 14553241481573, 18901300532988407, 40603763694792631, 9565202506169243753, 63888449105310899

Offset: 1

Alexander Adamchuk, May 15 2007

nonn

a(n) is a 3-d analog of Wolstenholme Numbers (A001008) that are the numerators of Harmonic Numbers H(n) = Sum[ 1/i, {i,1,n} ]. n X n X n cubic array M[i,j,k] = 1/(i+j+k-2) is a 3-d analog of n X n Hilbert Matrix with elements M[i,j] = 1/(i+j-1). p divides a((p+1)/3) for prime p = {5,11,17,23,29,41,47,53,59,71,83,89,...} = A007528 Primes of form 6n-1. Sum[ Sum[ Sum[ (i+j+k-2), {i,1,n} ], {j,1,n} ], {k,1,n} ] = 1/2*n^3*(3n-1).

Eric Weisstein, The World of Mathematics: Hilbert Matrix.
Eric Weisstein, The World of Mathematics: Harmonic Number.

Cf. A001008 = Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 1/i. Cf. A082687, A117731, A007528.

Table[ Numerator[ Sum[ Sum[ Sum[ 1/(i+j+k-2), {i,1,n} ], {j,1,n} ], {k,1,n} ] ], {n,1,30} ]

a(n) = Numerator[ Sum[ Sum[ Sum[ 1/(i+j+k-2), {i,1,n} ], {j,1,n} ], {k,1,n} ] ].

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

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A125741 The ratio of A117731(n) and A082687(n) when they are different.

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A119030 Difference between numerator and denominator of the sum of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n), A117731[n] - A117664[n].

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A126563 Numbers k such that the ratio of A117731(k) and A082687(k) is composite.

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A082687 Numerator of Sum_{k=1..n} 1/(n+k).

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A117664 Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.

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A296519 Denominator of n*Sum_{k=1..n} 1/(n+k).

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