cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A117731 Numerator of the fraction n*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, 54260455193, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193
Offset: 1

Views

Author

Alexander Adamchuk, Apr 14 2006

Keywords

Comments

a(n) almost always equals A082687(n), but differs for n in A125740.
p divides a((p-1)/3) for primes p in A002476, that is, primes of form 6*n + 1. - Alexander Adamchuk, Jul 16 2006

Examples

			The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/A296519.
For n=2, the n X n Hilbert matrix is
  1 1/2
  1/2 1/3
Thus, a(2) = numerator(1 + 1/2 + 1/2 + 1/3) = numerator(7/3) = 7.
The n X n Hilbert matrix begins as follows:
    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 ...
  ...

Links

Crossrefs

Cf. A296519 (denominators).
Cf. A001008, A002476, A005249, A058313, A058312, A082687, A086881, A098118, A117664, A125740.

Programs

  • Magma
    [Numerator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023
  • Mathematica
    Numerator[Table[n Sum[1/(n + k), {k, n}], {n, 1, 100}]]
Numerator[Table[Sum[Sum[1/(i + j - 1), {i, n}], {j, n}], {n, 30}]] (* Alexander Adamchuk, Apr 23 2006 *)
Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)
  • PARI
    a(n) = numerator(n*sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017
  • SageMath
    [numerator(n*(harmonic_number(2*n,1) - harmonic_number(n,1))) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

Formula

a(n) = numerator(n*Sum_{k=1..n} 1/(n+k)).
a(n) = numerator(n*(Psi(2*n+1) - Psi(n+1))).
a(n) = numerator(n*Sum_{k=1..2*n} (-1)^(k+1)/k).
a(n) = numerator(n*A058313(2*n)/A058312(2*n)).
a(n) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)), which is the 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). The denominator is A117664(n). - Alexander Adamchuk, Apr 23 2006

Extensions

Various sections edited by Petros Hadjicostas and Michel Marcus, May 07 2020

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

Views

Author

Alexander Adamchuk, Dec 04 2006

Keywords

Comments

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.

Examples

			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,...}.

Crossrefs

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} ].

Programs

  • Mathematica
    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} ]

Formula

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

Views

Author

Alexander Adamchuk, Mar 12 2007, Jun 09 2007

Keywords

Comments

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, ...}.

Links

Crossrefs

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

Programs

  • Mathematica
    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} ]
  • PARI
    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

Extensions

Edited by Max Alekseyev, Jul 12 2019
a(5)-a(7) from Jinyuan Wang, Jul 10 2025
Showing 1-3 of 3 results.

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