A086279 -id:A086279 - cp's OEIS frontend

A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

1, 2, 4, 3, 4, 8, 4, 4, 10, 8, 4, 12, 4, 8, 16, 5, 4, 20, 4, 12, 16, 8, 4, 16, 10, 8, 20, 12, 4, 32, 4, 6, 16, 8, 16, 30, 4, 8, 16, 16, 4, 32, 4, 12, 40, 8, 4, 20, 10, 20, 16, 12, 4, 40, 16, 16, 16, 8, 4, 48, 4, 8, 40, 7, 16, 32, 4, 12, 16, 32, 4, 40, 4, 8, 40, 12, 16, 32, 4, 20

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Dirichlet convolution of A001227 with itself.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A007426, A070288, A318366, A328487.

with(numtheory):
b:= proc(n) option remember; tau(2*n)-tau(n) end:
a:= n-> add(b(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 16 2019

nmax = 80; A001227 = Table[DivisorSum[n, Mod[#, 2] &], {n, 1, nmax}]; Table[DivisorSum[n, A001227[[#]] A001227[[n/#]] &], {n, 1, nmax}]
f[2, e_] := e + 1; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = Sum_{d|n} A001227(d) * A001227(n/d).
Sum_{k=1..n} a(k) ~ n * (log(n)^3/24 + (g/2 + log(2)/4 - 1/8)* log(n)^2 + (1/4 - g + 3*g^2/2 - log(2)/2 + 2*g*log(2) - sg1)* log(n) - 1/4 + (1 - 2*log(2))*g + (3*log(2) - 3/2)*g^2 + g^3 + log(2)/2 - log(2)^3/6 + (1 - 3*g - 2*log(2))* sg1 + sg2/2), where g is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(2^e) = e + 1, and a(p^e) = (e + 1)*(e + 2)*(e + 3)/6 for odd primes p. - Amiram Eldar, Nov 30 2020

A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

Original entry on oeis.org

1, 9, 6, 8, 9, 6, 9, 0, 8, 3, 9, 1, 0, 5, 2, 8, 5, 4, 6, 4, 6, 4, 8, 9, 1, 4, 5, 3, 7, 9, 6, 6, 8, 0, 5, 4, 2, 3, 1, 1, 3, 7, 7, 9, 4, 2, 8, 6, 8, 1, 9, 8, 1, 3, 4, 4, 5, 5, 1, 4, 3, 1, 5, 3, 4, 0, 2, 2, 5, 2, 1, 9, 8, 2, 6, 8, 9, 2, 3, 3, 4, 1, 1, 8, 6, 4, 4, 9, 1, 8, 3, 7, 4, 5, 7, 6, 7, 4, 4, 0, 9, 8, 7, 8, 3

Offset: 1

Amiram Eldar, Jun 09 2023

The formula for this sum was found by Olivier Oloa and proved by Roberto Tauraso in 2014.

			1.96896908391052854646489145379668054231137794286819...

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.
Olivier Oloa, A closed form of the series Sum_{n=1..oo} (H(n)^2 - (gamma + ln(n))^2)/n, Mathematics StackExchange, 2014.

Cf. A001008, A001620, A002117, A002805, A082633, A086279, A363538, A363540.

RealDigits[-StieltjesGamma[2] - 2*EulerGamma*StieltjesGamma[1] - 2*EulerGamma^3/3 + 5*Zeta[3]/3, 10, 120][[1]]

Equals -gamma_2 - 2*gamma*gamma_1 - (2/3)*gamma^3 + (5/3)*zeta(3), where gamma_1 and gamma_2 are the 1st and 2nd Stieltjes constants (A082633, A086279).

A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

Original entry on oeis.org

5, 8, 2, 1, 7, 4, 0, 0, 8, 5, 0, 4, 8, 6, 4, 6, 5, 2, 8, 8, 9, 6, 8, 6, 8, 6, 1, 5, 5, 0, 2, 0, 4, 1, 3, 4, 3, 1, 5, 0, 3, 3, 3, 2, 4, 3, 1, 9, 5, 7, 7, 0, 1, 1, 4, 4, 2, 4, 0, 3, 9, 2, 7, 6, 4, 7, 6, 4, 1, 3, 9, 7, 2, 2, 5, 9, 8, 1, 8, 9, 7, 4, 9, 5, 1, 8, 9, 0, 4, 2, 8, 5, 7, 4, 3, 2, 3, 1, 9, 0, 9, 6, 5, 9, 7

Offset: 1

Amiram Eldar, Jun 09 2023

			5.82174008504864652889686861550204134315033324319577...

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.

Cf. A001008, A001620, A002117, A013662, A082633, A086279, A086280, A363538, A363539.

RealDigits[-StieltjesGamma[3] - 3*EulerGamma*StieltjesGamma[2] - 3*EulerGamma^2*StieltjesGamma[1] - 3*EulerGamma^4/4 + 43*Zeta[4]/8, 10, 120][[1]]

Equals -gamma_3 - 3*gamma*gamma_2 - 3*gamma^2*gamma_1 - (3/4)*gamma^4 + (43/8)*zeta(4), where gamma_1, gamma_2 and gamma_3 are the 1st, 2nd and 3rd Stieltjes constants (A082633, A086279, A086280).

A368568 Decimal expansion of the Wolf-Kawalec constant of index 2.

Original entry on oeis.org

3, 1, 9, 3, 8, 4, 1, 2, 0, 4, 0, 8, 0, 1, 4, 2, 4, 9, 2, 4, 9, 4, 6, 5, 2, 0, 7, 0, 7, 4, 5, 7, 2, 0, 1, 5, 2, 8, 1, 6, 1, 4, 2, 9, 2, 0, 2, 4, 7, 8, 3, 7, 2, 3, 8, 7, 0, 0, 2, 3, 0, 4, 9, 0, 5, 6, 0, 1, 4, 9, 0, 5, 6, 8, 4, 2, 6, 7, 7, 1, 3, 4, 1, 4, 6, 9, 7, 4, 3, 2, 4, 1, 1, 1, 4, 4, 5, 1, 9, 0, 6, 0, 2, 6, 5

Offset: 0

Artur Jasinski, Dec 30 2023

For the Wolf-Kawalec constant of index 0 see A368551.

For the Wolf-Kawalec constant of index 1 see A368547.

			0.3193841204080142492494652...

Artur Kawalec, On the series expansion of a square-free zeta series, arXiv:2312.16811 [math.NT], 2023 see Table 1.
Marek Wolf, Numerical Determination of a Certain Mathematical Constant Related to the Mobius Function, Computational Methods in Science and Technology, Volume 29 (1-4) 2023, 17-20 see formula (20).

Cf. A001620, A073002, A086279, A201994, A368547, A368551.

RealDigits[Limit[D[D[Zeta[x]/Zeta[2 x] - 6/(Pi^2 (x - 1)), x], x], x -> 1],10,105][[1]]

Equals 6*(Pi^6*gamma_2 - 3456*(zeta'(2))^3 + 288*Pi^2*zeta'(2)*(gamma*zeta'(2) + 2*zeta''(2)) + 8*Pi^4*(3*gamma_1*zeta(2) - 3*gamma*zeta''(2) - 2*zeta'''(2)))/Pi^8 where gamma_2 is A086279.

A082632 Decimal expansion of lim_{y->infinity} y^2*(Re(zeta(1 + i/y)) - gamma).

Original entry on oeis.org

0, 0, 4, 8, 4, 5, 1, 8, 1, 5, 9, 6, 4, 3, 6, 1, 5, 9, 2, 4, 2, 2, 6, 5, 1, 9, 3, 0, 1, 7, 6, 0, 6, 2, 6, 4, 6, 7, 9, 5, 3, 2, 9, 0, 3, 0, 5, 0, 6, 7, 0, 3, 7, 4, 9, 4, 0, 3, 5, 0, 6, 8, 2, 7, 2, 5, 9, 2, 5, 3, 7, 7, 6, 9, 1, 1, 4, 0, 2, 0, 7, 0, 8, 5, 9, 8, 9, 0, 9, 8, 6, 9, 0, 6, 8, 7, 2, 6, 8, 6, 5, 9, 6, 4, 3

Offset: 0

Benoit Cloitre, May 24 2003

Lim_{y->infinity} Im(zeta(1 + i/y))/y = -1. - Vaclav Kotesovec, Feb 18 2021

			0.00484518159643615924226519301760626...

Cf. A086279.

StieltjesGamma[2]/2 // RealDigits[#, 10, 103]& // First // Join[{0, 0}, #]& (* Jean-François Alcover, Mar 04 2013 *)

Equals lim_{y->infinity} y^2*(Re(zeta(1+i/y)) - gamma), where gamma is the Euler-Mascheroni constant A001620.

Equals A086279 / 2. - R. J. Mathar, Jul 15 2010

More terms from Jean-François Alcover, Mar 04 2013

A385612 Decimal expansion zeta''''(0) (negated).

Original entry on oeis.org

2, 3, 9, 9, 7, 1, 0, 3, 1, 8, 8, 0, 1, 3, 7, 0, 7, 9, 5, 8, 9, 8, 7, 2, 1, 9, 5, 2, 7, 7, 4, 1, 0, 0, 5, 6, 6, 1, 8, 9, 1, 1, 3, 9, 9, 3, 4, 9, 2, 1, 7, 0, 3, 4, 2, 4, 9, 7, 6, 0, 0, 9, 3, 3, 3, 0, 4, 6, 3, 8, 2, 9, 3, 8, 6, 3, 3, 4, 4, 9, 9, 1, 3, 8, 2, 8, 6, 1, 8, 2, 2, 7, 5, 7, 8, 1, 3, 3, 4, 6, 9, 4, 9, 0, 3

Offset: 2

Artur Jasinski, Jul 04 2025

n-th derivative of zeta function at 0 is close to -n!, which here is the present constant close to 4! = 24.

			23.997103188013707958987219527741...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.

Cf. A075700, A257549, A261508.

Cf. A001620, A061444, A082633, A086279, A086280.

evalf(-Zeta(4, 0), 120); # Vaclav Kotesovec, Jul 04 2025

RealDigits[-3 EulerGamma^4/2 - EulerGamma^2 Pi^2/4 + 19 Pi^4/480 - 4 EulerGamma^3 Log[2 Pi] - 3 EulerGamma^2 Log[2Pi]^2 +  Pi^2 Log[2 Pi]^2/4 + Log[2 Pi]^4/2 - 6 EulerGamma^2 StieltjesGamma[1] - Pi^2 StieltjesGamma[1]/2 - 12 EulerGamma Log[2 Pi] StieltjesGamma[1] - 6 Log[2 Pi]^2 StieltjesGamma[1] - 6 EulerGamma StieltjesGamma[2] - 6 Log[2Pi] StieltjesGamma[2] - 2 StieltjesGamma[3] + 4 Log[2 Pi] Zeta[3],10,105][[1]]

PARI
```
-zeta''''(0)
```

Equals -3*gamma^4/2 - gamma^2*Pi^2/4 + 19*Pi^4/480 - 4*gamma^3*log(2*Pi) -3*gamma^2*log(2*Pi)^2 + Pi^2*log(2*Pi)^2/4 + log(2*Pi)^4/2 - 6*gamma^2*StieltjesGamma(1) - Pi^2*StieltjesGamma(1)/2 - 12*gamma*log(2*Pi)* StieltjesGamma(1) - 6*log(2*Pi)^2*StieltjesGamma(1) - 6*gamma*StieltjesGamma(2) - 6*log(2*Pi)*StieltjesGamma(2) - 2*StieltjesGamma(3) + 4*log(2*Pi)*zeta(3).

A386718 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(xyz)} dx dy dz, where {} denotes fractional part.

Original entry on oeis.org

5, 0, 0, 4, 4, 5, 3, 6, 2, 1, 7, 8, 5, 8, 0, 0, 2, 3, 4, 9, 6, 3, 3, 9, 4, 7, 8, 8, 1, 0, 1, 0, 5, 1, 5, 2, 7, 7, 5, 1, 0, 9, 9, 0, 5, 4, 4, 5, 0, 8, 4, 7, 2, 8, 7, 3, 3, 5, 9, 0, 0, 0, 7, 5, 8, 2, 4, 5, 9, 0, 8, 4, 4, 8, 4, 9, 8, 7, 0, 2, 1, 0, 2, 7, 1, 2, 8, 9, 6, 3, 6, 4, 3, 7, 8, 4, 5, 3, 3, 7, 4, 9, 0, 8, 8

Offset: 0

Amiram Eldar, Jul 31 2025

			0.50044536217858002349633947881010515277510990544508...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.43, page 106.

Yaming Yu, A Multiple Integral in Terms of Stieltjes Constants, SIAM Problems and Solutions, Classical Analysis, Integrals, Problem 07-002 (2007).

Cf. A001620 (gamma), A082633 (-gamma_1), A086279 (-gamma_2).

Cf. A153810 (m=1), A242610 (m=2), this constant (m=3).

With[{m = 2}, RealDigits[1 - Sum[StieltjesGamma[k]/k!, {k, 0, 2}], 10, 120][[1]]]

Equals 1 - gamma - gamma_1 - gamma_2/2, where gamma_k is the k-th Stieltjes constant.

In general, for m >= 1, Integral_{x_1=0..1} ... Integral_{x_m=0..1} {1/(x_1*...*x_m)} dx_1 ... dx_m = 1 - Sum_{k=0..m-1} gamma_k/k!, where gamma_0 = gamma is Euler's constant.

cp's OEIS Frontend

A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

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A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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A368568 Decimal expansion of the Wolf-Kawalec constant of index 2.

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A082632 Decimal expansion of lim_{y->infinity} y^2*(Re(zeta(1 + i/y)) - gamma).

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A385612 Decimal expansion zeta''''(0) (negated).

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A386718 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x*y*z)} dx dy dz, where {} denotes fractional part.

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A386718 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(xyz)} dx dy dz, where {} denotes fractional part.