A358625 -id:A358625 - cp's OEIS frontend

A120082 Numerators of expansion for Debye function for n=1: D(1,x).

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120083.
D(1,x) = (1/x)*integral_{t=0..x} t/(exp(t)-1) dt (note the factor x on the r.h.s. of  the Abramowitz-Stegun link). This is the e.g.f. for {Bernoulli(n)/(n+1)}A027641(n)/A227540(n).%20Thanks%20to%20_Peter%20Luschny">{n>=0}. See A027641(n)/A227540(n). Thanks to _Peter Luschny for asking me to revisit this sequence. - Wolfdieter Lang, Jul 15 2013
Also numerators of coefficients in expansion of x/(exp(x)-1). See A227830 for denominators. - N. J. A. Sloane, Aug 01 2013

			Rationals r(n): [1, -1/4, 1/36, 0, -1/3600, 0, 1/211680, 0, -1/10886400, ...].

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 23.

Peter Luschny, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=1, with a factor x extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A060054, A060055, A120083, A182918, A227540, A227830, A358625.

[Numerator(Bernoulli(n)/Factorial(n+1)): n in [0..50]]; // G. C. Greubel, May 01 2023

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: # R. J. Mathar, Sep 03 2009
gf := (1 - x/4 + sum((bernoulli(2*k)/((2*k+1)*(2*k)!))*x^(2*k), k=0..infinity)):
a := proc(n) local ser; if n = 0 then return 1 fi; ser := series(gf, x, n+2):
numer(coeff(ser, x, n)) end: seq(a(n), n = 0..40); # Peter Luschny, Dec 02 2022

Table[Numerator[BernoulliB[n]/((n+1)!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

def A120082(n): return numerator(bernoulli(n)/factorial(n+1))
[A120082(n) for n in range(51)] # G. C. Greubel, May 01 2023

a(n) = numerator(r(n)), with r(n) = [x^n] (1 - x/4 + Sum_{k>=0} (B(2*k)/((2*k+1)*(2*k)!))*x^(2*k)), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).
a(n) = numerator(B(n)/(n+1)!), n >= 0. See the above comment on the e.g.f. D(1,x). - Wolfdieter Lang, Jul 15 2013
Apart from the sign of a(1) this sequence differs from A358625 for the first time at n = 68. - Peter Luschny, Dec 02 2022

Edited after Andrey Zabolotskiy noticed an inconsistency by Peter Luschny, Dec 02 2022

A362870 a(n) = sigma_29(n), the sum of the 29th powers of the divisors of n.

Original entry on oeis.org

1, 536870913, 68630377364884, 288230376688582657, 186264514923095703126, 36845653355419807219092, 3219905755813179726837608, 154742505198902911050973185, 4710128697246313465298968573, 100000000186264514923632574038, 1586309297171491574414436704892

Offset: 1

Vaclav Kotesovec, May 07 2023

In general, for k > 0, Sum_{n>=1} sigma_(4*k+1)(n) / exp(2*Pi*n) = Bernoulli(4*k+2)/(8*k+4). For k = 0, Sum_{n>=1} sigma(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = Bernoulli(2)/4 - 1/(8*Pi).

This formula can best be understood as a statement about the divided Bernoulli numbers b(n) = B(n) / n. Then you can say: If v is twice an odd number greater than 1 (i.e., v = 4*n + 2, a term of A016825 that is greater than 2), then b(v) = 2 * Sum_{j>=1} sigma_{v - 1}(j) / exp(2*Pi*j) = A358625(v) / A075180(v - 1). - Peter Luschny, May 08 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Luschny, Is this a new representation of (some) Bernoulli numbers?, Mathematics Stack Exchange.
Index entries for sequences related to sigma(n).

Cf. A000203 (sigma_1), A001160 (sigma_5), A013957 (sigma_9), A013961 (sigma_13), A013965 (sigma_17), A013969 (sigma_21), A281959 (sigma_25).

Cf. A075180, A358625, A016825, A082771.

with(NumberTheory): seq(SumOfDivisors(k, 29), k = 1..20);

Mathematica
```
DivisorSigma[29, Range[20]]
```

for(n=1, 20, print1(direuler( p=2, n, 1 / (1 - X) /(1 - p^29*X))[n], ", "))

from sympy import divisor_sigma
def A362870(n): return divisor_sigma(n,29) # Chai Wah Wu, May 07 2023

G.f.: Sum_{k>=1} k^29 * x^k / (1-x^k).
Dirichlet g.f.: zeta(s-29)*zeta(s).
Sum_{k=1..n} a(k) ~ zeta(30) * n^30 / 30.
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1723168255201/171864 = Bernoulli(30)/60.
Multiplicative with a(p^e) = (p^(29*e+29)-1)/(p^29-1). - Amiram Eldar, Oct 29 2023

cp's OEIS Frontend

A120082 Numerators of expansion for Debye function for n=1: D(1,x).

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