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 1-4 of 4 results.

A075180 Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

Original entry on oeis.org

2, 12, 1, 120, 1, 252, 1, 240, 1, 132, 1, 32760, 1, 12, 1, 8160, 1, 14364, 1, 6600, 1, 276, 1, 65520, 1, 12, 1, 3480, 1, 85932, 1, 16320, 1, 12, 1, 69090840, 1, 12, 1, 541200, 1, 75852, 1, 2760, 1, 564, 1, 2227680, 1, 132, 1, 6360, 1, 43092, 1, 6960, 1, 708, 1, 3407203800, 1, 12, 1, 32640, 1, 388332, 1, 120, 1, 9372, 1, 10087262640, 1, 12
Offset: 0

Views

Author

Wolfdieter Lang, Sep 06 2002

Keywords

Comments

Denominators of -zeta(-n), n >= 0, where zeta is Riemann's zeta function.
Numerators are +1, A060054(n+1), n >= 1.

Examples

			1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, 0, -691/32760, ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Links

Crossrefs

Cf. A060054, A006232 with A006233.
Cf. A006953, A006863, A079612, A242179, A106831, A000142, A300711.

Programs

  • Haskell
    a075180 n = a075180_list !! n
a075180_list = map (denominator . sum) $ zipWith (zipWith (%))
   (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf
-- Reinhard Zumkeller, Jul 04 2014
  • Maple
    a := n -> denom(bernoulli(n+1,1)/(n+1)); # Peter Luschny, Apr 22 2009
  • Mathematica
    a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Denominator[a[i]], {i,0,20}] (* Peter Luschny, Apr 29 2009 *)
Table[Denominator[Zeta[-n]], {n, 0, 49}] (* Alonso del Arte, Jan 13 2012 *)
CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 48}], 1/n] // Rest // Denominator (* Jean-François Alcover, Mar 28 2013 *)
With[{nn=50},Denominator[CoefficientList[Series[1/(1-Exp[-x])-1/x,{x,0,nn}],x] Range[0,nn-1]!]] (* Harvey P. Dale, Apr 13 2016 *)
  • PARI
    x='x+O('x^66);
egf = 1/(1-exp(-x)) - 1/x;
v=Vec(serlaplace(egf));
vector(#v,n, denominator(v[n]))
/* Joerg Arndt, Mar 28 2013 */
  • PARI
    A075180(n) = denominator(bernfrac(n+1)/(n+1)); \\ Antti Karttunen, Dec 19 2018, after Maple-program.

Formula

a(n) = denominator(-Zeta(-n)) = denominator(((-1)^(n+1))*B(n+1)/(n+1)), n >= 0, with Riemann's zeta function and the Bernoulli numbers B(n).
a(n) = denominators from e.g.f. (B(-x) - 1)/x, with B(x) = x/(exp(x) - 1), e.g.f. for Bernoulli numbers A027641(n)/A027642(n), n >= 0.
From Jianing Song, Apr 05 2021: (Start)
a(2n-1) = A006863(n)/2 for n > 0. By the comments in A006863, A006863(n) = A079612(2n) for n > 0. Hence a(n) = A079612(n+1)/2 all odd n. For all even n > 0, we have a(n) = 1, which is also equal to A079612(n+1)/2.
For odd n, a(n) is the product of p^(e+1) where p^e*(p-1) divides n+1 but p^(e+1)*(p-1) does not. For example, a(11) = 2^3 * 3^2 * 5^1 * 7^1 * 13^1 = 32760.
a(2n-1) = A002445(n)*(2n)/A300711(n), n > 0. (End)
a(2*n-1) = A006953(n) for n >= 1. - Georg Fischer, Dec 01 2022

Extensions

More terms from Antti Karttunen, Dec 19 2018

A111008 a(n) = A000367(n)/A141590(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 629, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 1003, 481, 53, 1, 5, 7, 19, 29, 59, 1, 61, 2077, 103, 1, 65, 11, 67, 17, 23, 259, 71, 1, 73, 37, 25, 2489, 77
Offset: 0

Views

Author

Paul Curtz, Aug 25 2008

Keywords

Links

Crossrefs

See A141517.
Cf. A300711.

Programs

  • Maple
    A120082 := proc(n) local b; if n = 0 then b := 1 ; elif n = 1 then b := -1/4 ; elif type(n,'odd') then b := 0; else b := bernoulli(n)/(n+1)! ; fi; numer(b) ; end:
A141590 := proc(n) A120082(2*n) ; end: A000367 := proc(n) numer(bernoulli(2*n)) ; end:
A111008 := proc(n) A000367(n)/A141590(n) ; end: seq(A111008(n),n=0..120) ; # R. J. Mathar, Sep 03 2009
  • PARI
    upto(N)=bernvec(N);forstep(n=0,2*N,2,print1(gcd(numerator(bernfrac(n)), (n+1)!),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

Extensions

Edited and extended by R. J. Mathar, Sep 03 2009

A195989 Quotient of denominators of (BernoulliB(2n)/n) and BernoulliB(2n).

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 2, 3, 16, 1, 18, 1, 20, 21, 2, 1, 24, 1, 2, 27, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 40, 1, 42, 1, 4, 9, 2, 1, 48, 1, 50, 3, 4, 1, 54, 11, 8, 3, 2, 1, 60, 1, 2, 63, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 81, 2, 1, 84
Offset: 1

Views

Author

Paul Curtz, Dec 21 2012

Keywords

Comments

The fixed points (entries equal to their index) are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 21, 24, 27, 30, 32, 36, 40, 42,... See A193267.
Are the indices of the 1's, that is 1, 5, 7, 11, 13,... , the sequence A069040 (checked to be true for their first 700 entries)? This provides another link between the Bernoulli numbers.
a(10*k) = 10, 20, 30, 40, 50, 60, 10, 70, 80, 90, 100,... for k= 1, 2, 3,....

Examples

			a(1) = 6/6 =1, a(2) = 60/30 =2, a(3) =126/42 =3, a(4) = 120/30 =4, a(5) = 66/66 =1.

Links

Programs

  • Magma
    [Denominator(Bernoulli(2*n)/n)/Denominator(Bernoulli(2*n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018
  • Maple
    A195989 := proc(n)
    q1 := denom(bernoulli(2*n)/n) ;
    q2 := denom(bernoulli(2*n)) ;
    q1/q2 ;
end proc: # R. J. Mathar, Jan 06 2013
# Alternatively, without Bernoulli numbers:
A195989 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; n/P end: seq(A195989(n),n=1..84); # Peter Luschny, Mar 12 2018
  • Mathematica
    a[n_] := Denominator[ BernoulliB[2*n]/n] / Denominator[ BernoulliB[2*n]]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Jan 04 2013 *)
  • PARI
    a(n) = my(b=bernfrac(2*n)); denominator(b/n)/denominator(b); \\ Michel Marcus, Mar 12 2018

Formula

a(n) = A193267(2*n)/2 = A036283(n) / A002445(n).
a(n) = n/A300711(n). - Peter Luschny, Mar 12 2018
2a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 divides 2n. - Peter Luschny, Mar 12 2018

A300330 a(n) is the product over all prime powers p^e where p^e is the highest power of p dividing n and p-1 does not divide n.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 1, 19, 1, 21, 11, 23, 1, 25, 13, 27, 7, 29, 5, 31, 1, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 1, 55, 7, 57, 29, 59, 1, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 1
Offset: 1

Views

Author

Peter Luschny, Mar 12 2018

Keywords

Links

Crossrefs

Cf. A193267, A195989, A300711.

Programs

  • Julia
    using Nemo
function A300330(n) P = 1
    for (p, e) in factor(ZZ(n))
        ! divisible(ZZ(n), p - 1) && (P *= p^e) end
P end
[A300330(n) for n in 1:72] |> println
  • Magma
    [n/(Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n))): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018
  • Maple
    A300330 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(n)[2]; for f in F do if not divides(f[1]-1, n) then
P := P*f[1]^f[2] fi od; P end: seq(A300330(n), n=1..100);
  • Mathematica
    a[n_]:=If[OddQ[n], 1, Denominator[BernoulliB[n]/n]/Denominator[BernoulliB[n]]]; Table[n/a[n], {n, 1, 100}] (* Vincenzo Librandi, Mar 12 2018 *)

Formula

a(n) * A193267(n) = n.
Showing 1-4 of 4 results.

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