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.

A119682 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^2.

Original entry on oeis.org

1, 3, 31, 115, 3019, 973, 48877, 191833, 5257891, 5194387, 634871227, 629535127, 107159834863, 106497287263, 107074439839, 426268707331, 123711093737059, 41082589491553, 14880853934789833, 2967138724292741, 2975331071381381
Offset: 1

Views

Author

Alexander Adamchuk, Jun 08 2006, Jun 25 2006

Keywords

Comments

p divides a(p-1) for prime p > 2 -- similar to Wolstenholme's theorem for A007406(n) (numerator of Sum_{k=1..n} 1/k^2).

Examples

			The first few fractions are 1, 3/4, 31/36, 115/144, 3019/3600, 973/1200, 48877/58800, 191833/235200, 5257891/6350400, ... = A119682/A334580. - _Petros Hadjicostas_, May 06 2020

Links

Crossrefs

Cf. A003418, A007406, A334580 (denominators).

Programs

  • GAP
    List(List([1..25],n->Sum([1..n],k->(-1)^(k+1)*(1/k^2))),NumeratorRat); # Muniru A Asiru, Apr 07 2018
  • Maple
    seq(numer(simplify(LerchPhi(-1,2,n)*(-1)^n+Pi^2/12-(-1)^n/n^2)),n=1..30); # Robert Israel, May 30 2018
  • Mathematica
    Numerator[Table[Sum[(-1)^(i+1)*1/i^2,{i,1, n}],{n,1,40}]]
Sqrt[Numerator[Table[Sum[Sum[(-1)^(i+j)*1/(i*j)^2, {i, 1, n}], {j, 1, n}],{n,1,20}]]] (* Alexander Adamchuk, Jun 26 2006 *)
a[n_] := 1/12 (Pi^2 - 3 (-1)^n Zeta[2, (1 + n)/2, IncludeSingularTerm -> False] + 3 (-1)^n Zeta[2, 1 + n/2, IncludeSingularTerm -> False]) // Simplify // Numerator
Table[a[n], {n, 1, 22}]  (* Gerry Martens, Jun 01 2018 *)
  • PARI
    a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^2)); \\ Altug Alkan, Apr 06 2018
  • PARI
    first(n) = {my(res = vector(n), s = 1); res[1] = 1; for(k = 2, n, s = -s; res[k] = res[k - 1] + s/k^2; res[k - 1] = numerator(res[k - 1])); res} \\ David A. Corneth, Apr 07 2018

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/k^2).
a(n) = abs(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*j/i^2)). - Alexander Adamchuk, Jun 26 2006
a(n) = sqrt(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)/(i*j)^2)). - Alexander Adamchuk, Jun 26 2006
a(n) = numerator((1/12)*(Pi^2-3*(-1)^n*(zeta(2,(1+n)/2)-zeta(2,(2+n)/2)))). - Gerry Martens, Apr 07 2018

A068589 Denominator(Sum_{i=1..n} 1/i^3)/denominator(Sum_{i=1..n} 1/i).

Original entry on oeis.org

1, 4, 36, 144, 3600, 1200, 58800, 235200, 6350400, 6350400, 768398400, 768398400, 129859329600, 129859329600, 129859329600, 519437318400, 150117385017600, 50039128339200, 18064125330451200, 18064125330451200, 54192375991353600
Offset: 1

Views

Author

Benoit Cloitre, Mar 27 2002

Keywords

Comments

For n = 1 to n = 19, we have a(n) = A334580(n), but a(20) = 18064125330451200 <> 3612825066090240 = A334580(20). - Petros Hadjicostas, May 06 2020

Crossrefs

Cf. A002805, A007409, A119682, A334580.

Programs

  • Maple
    a := proc(n) local i: denom(add(1/i^3, i = 1 .. n))/denom(add(1/i, i = 1 .. n)): end proc:
seq(a(n), n=1..30); # Petros Hadjicostas, May 06 2020
  • PARI
    a(n) = denominator(sum(k=1, n, 1/k^3)/sum(k=1, n, 1/k)); \\ Michel Marcus, May 07 2020

Formula

a(n) = A007409(n)/A002805(n).

A357556 a(n) is the denominator of Sum_{d|n} (-1)^(d+1) / d^2.

Original entry on oeis.org

1, 4, 9, 16, 25, 6, 49, 64, 81, 50, 121, 72, 169, 98, 45, 256, 289, 108, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 15, 961, 1024, 1089, 578, 49, 1296, 1369, 722, 1521, 800, 1681, 147, 1849, 88, 2025, 1058, 2209, 128, 2401, 2500, 2601, 1352, 2809, 243, 3025
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 03 2022

Keywords

Examples

			1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Crossrefs

Cf. A017668, A064027, A098988, A321543, A334580, A357555 (numerators).

Programs

  • Mathematica
    Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Denominator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
  • PARI
    a(n) = denominator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022
  • Python
    from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).denominator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

Formula

Denominators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).
Showing 1-3 of 3 results.

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