[go: nahoru, domu]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a340839 -id:a340839
Displaying 1-7 of 7 results found. page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1). +10
15
1, 0, 1, 0, 9, 1, 5, 1, 6, 0, 6, 0, 1, 0, 1, 9, 5, 2, 2, 6, 0, 4, 9, 5, 6, 5, 8, 4, 2, 8, 9, 5, 1, 4, 9, 2, 0, 9, 8, 4, 5, 3, 8, 6, 2, 7, 5, 8, 1, 7, 3, 8, 5, 2, 3, 7, 3, 2, 0, 2, 4, 2, 0, 0, 8, 9, 2, 5, 1, 6, 1, 3, 7, 4, 2, 4, 5, 6, 7, 2, 6, 3, 7, 0, 9, 3, 9, 6, 1, 9, 7, 6, 9, 4, 5, 5, 8, 9, 2, 1, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
This constant is called Euler product 2==1 modulo 5 (see Mathar's Definition 5 formula (38)) or equivalently zeta 2==1 modulo 5.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..501
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 p.20 (100 digits precision data).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2014-2015, Section 3.3. zeta_{5,1}(2).
FORMULA
Equals Sum_{k>=1} 1/A004615(k)^2. - Amiram Eldar, Jan 24 2021
Equals exp(-gamma/2)*Pi/(A340839^2*sqrt(5*log((1 + sqrt (5))/2))). - Artur Jasinski, Jan 30 2021
EXAMPLE
1.01091516060101952260495658428951492...
MATHEMATICA
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[5, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021, took 20 minutes *)
CROSSREFS
Cf. A340127, A340628, A340629, A340665, A340794, A340839.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A004615, A340576, A340577, A340578, A334826.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 15 2021
STATUS
approved
A340866 Decimal expansion of the Mertens constant C(5,4). +10
5
1, 2, 9, 9, 3, 6, 4, 5, 4, 7, 9, 1, 4, 9, 7, 7, 9, 8, 8, 1, 6, 0, 8, 4, 0, 0, 1, 4, 9, 6, 4, 2, 6, 5, 9, 0, 9, 5, 0, 2, 5, 7, 4, 9, 7, 0, 4, 0, 8, 3, 2, 9, 6, 6, 2, 0, 1, 6, 7, 8, 1, 7, 7, 0, 3, 1, 2, 9, 2, 2, 8, 7, 8, 8, 3, 5, 4, 4, 0, 3, 5, 8, 0, 6, 4, 7, 6, 4, 7, 6, 9, 7, 6, 7, 6, 5, 7, 9, 3, 0, 2, 9, 4, 0, 9, 3, 5, 5, 0, 7, 6, 3, 7, 3, 7, 4, 3, 2, 1, 5, 4, 2, 7, 1, 1, 9, 0, 7, 0, 3, 3, 5, 4, 0, 9, 8, 6, 0, 6, 1, 4, 5, 0, 3, 2, 9, 7, 2, 5, 8, 8, 4, 3, 6, 1, 1, 5, 9, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Data taken from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..497
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007). [in this table on page 4, the last correct digit is a(108), beyond the level there certified. - Vaclav Kotesovec, Jan 26 2021]
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
For other links see A340711.
FORMULA
Equals A340839*5^(1/4)*sqrt(A340004/(2*A340127)).
Equals (13*Pi^2/(24*sqrt(5)*exp(gamma)*log((1+sqrt(5))/2))*A340629/A340809)^(1/4). - Vaclav Kotesovec, Jan 25 2021
EXAMPLE
1.299364547914977988160840014964265909502574970408329662016...
MATHEMATICA
(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[(13*Pi^2 / (24*Sqrt[5] * Exp[EulerGamma] * Log[(1 + Sqrt[5])/2]) * Z[5, 1, 2]^2 / (Z[5, 1, 4] * Z[5, 4, 4]))^(1/4)]
CROSSREFS
Cf. A340004, A340127, A340839, A340711.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 24 2021
EXTENSIONS
Corrected by Vaclav Kotesovec, Jan 25 2021
More digits from Vaclav Kotesovec, Jan 26 2021
STATUS
approved
A340857 Decimal expansion of constant K5 = 29*log(2+sqrt(5))*(Product_{primes p == 1 (mod 5)} (1-4*(2*p-1)/(p*(p+1)^2)))/(15*Pi^2). +10
2
2, 6, 2, 6, 5, 2, 1, 8, 8, 7, 2, 0, 5, 3, 6, 7, 6, 6, 6, 7, 5, 9, 6, 2, 0, 1, 1, 4, 7, 2, 0, 8, 8, 3, 4, 6, 5, 3, 0, 2, 0, 4, 3, 9, 3, 0, 6, 4, 7, 4, 4, 7, 3, 9, 1, 0, 6, 8, 2, 5, 5, 1, 0, 5, 8, 7, 0, 9, 2, 6, 6, 8, 3, 8, 6, 9, 0, 2, 2, 7, 4, 1, 7, 9, 4, 1, 9, 3, 8, 3, 6, 5, 5, 2, 3, 5, 0, 0, 2, 0, 1, 0, 0, 8, 9, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Finch and Sebah, 2009, p. 7 (see link) call this constant K_5. K_5 is related to the Mertens constant C(5,1) (see A340839). For more references see the links in A340711. Finch and Sebah give the following definition:
Consider the asymptotic enumeration of m-th order primitive Dirichlet characters mod n. Let b_m(n) denote the count of such characters. There exists a constant 0 < K_m < oo such that Sum_{n <= N} b_m(n) ∼ K_m*N*log(N)^(d(m) - 2) as N -> oo, where d(m) is the number of divisors of m.
LINKS
Table of n, a(n) for n=0..105.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 p. 10.
FORMULA
Equals (29/25)*(Product_{primes p} (1-1/p)^2*(1+gcd(p-1,5)/(p-1))) [Finch and Sebah, 2009, p. 10].
EXAMPLE
0.262652188720536766675962011472088346530204393064744739106825510587...
MATHEMATICA
$MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 4*(2*p-1)/(p*(p+1)^2));
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]]*S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[5, 1, m]; sump = sump + difp; PrintTemporary[m]; m++];
RealDigits[Chop[N[29*Log[2+Sqrt[5]]/(15*Pi^2) * Exp[sump], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 25 2021, took over 50 minutes *)
CROSSREFS
Cf. A340878 (K3), A340879 (K4).
Cf. A340004, A340794, A340665, A340127.
Cf. A340629, A340710, A340711, A340628.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A340576, A340577, A340578, A334826.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 24 2021
STATUS
approved
A335576 Decimal expansion of Mertens constant C(5,2). +10
1
5, 4, 6, 9, 7, 5, 8, 4, 5, 4, 1, 1, 2, 6, 3, 4, 8, 0, 2, 3, 8, 3, 0, 1, 2, 8, 7, 4, 3, 0, 8, 1, 4, 0, 3, 7, 7, 5, 1, 9, 9, 6, 3, 2, 4, 1, 0, 0, 8, 1, 9, 2, 9, 5, 1, 5, 3, 1, 2, 7, 1, 8, 7, 1, 9, 1, 7, 5, 1, 8, 1, 1, 0, 8, 5, 7, 1, 5, 1, 6, 6, 8, 3, 3, 5, 8, 4, 0, 6, 3, 7, 2, 3, 8, 3, 5, 4, 8, 2, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
First 100 digits from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.
LINKS
Table of n, a(n) for n=0..99.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134.
FORMULA
A = C(5,1)=1.2252384385390845800576097747492205... see A340839.
B = C(5,2)=0.5469758454112634802383012874308140... this constant.
C = C(5,3)=0.8059510404482678640573768602784309... see A336798.
D = C(5,4)=1.2993645479149779881608400149642659... see A340866.
A*B*C*D = 0.70182435445860646228... = (5/4)*exp(-gamma), where gamma is the Euler-Mascheroni constant A001620.
B = sqrt(2)*5^(3/4)*sqrt(A340127)*exp(-gamma)/(4*sqrt(A340004)*A^2*C).
B = 2*A*D*log((1+sqrt(5))/2)/(C*sqrt(5)*A340794*A340665).
B = A*D*log((1+sqrt(5))/2)^2/(C*Pi*A340213^2).
From Vaclav Kotesovec, Jan 27 2021: (Start)
B*C = 5^(1/4) * exp(-gamma/2) * sqrt(log((1+sqrt(5))/2) / (2 * A340665 * A340794)).
A*D = 5^(3/4) * exp(-gamma/2) * sqrt(A340665 * A340794 / (8 * log((1+sqrt(5))/2))).
(End)
EXAMPLE
0.546975845411263480238301287430814...
CROSSREFS
Cf. A001620, A336798, A340004, A340127, A340213, A340665, A340794, A340839, A340866.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 26 2021
STATUS
approved
A336798 Decimal expansion of Mertens constant C(5,3). +10
1
8, 0, 5, 9, 5, 1, 0, 4, 0, 4, 4, 8, 2, 6, 7, 8, 6, 4, 0, 5, 7, 3, 7, 6, 8, 6, 0, 2, 7, 8, 4, 3, 0, 9, 3, 2, 0, 8, 1, 2, 8, 8, 1, 1, 4, 9, 3, 9, 0, 1, 0, 8, 9, 7, 9, 3, 4, 8, 1, 6, 9, 4, 1, 2, 5, 2, 0, 7, 7, 6, 6, 1, 8, 8, 2, 6, 9, 8, 5, 5, 1, 3, 1, 1, 1, 9, 0, 1, 4, 4, 6, 8, 1, 0, 8, 5, 2, 6, 7, 9, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
First 101 digits from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.
A = C(5,1)=1.2252384385390845800576097747492205... see A340839.
B = C(5,2)=0.5469758454112634802383012874308140... see A335576.
C = C(5,3)=0.8059510404482678640573768602784309... this constant.
D = C(5,4)=1.2993645479149779881608400149642659... see A340866.
LINKS
Table of n, a(n) for n=0..100.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134.
FORMULA
For formulas see A335576.
EXAMPLE
0.80595104044826786405737686...
CROSSREFS
Cf. A335576, A340839, A340866.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 27 2021
STATUS
approved
A340213 Decimal expansion of the constant kappa(-5) = (1/2)*sqrt(sqrt(5)*log(9+4*sqrt(5))/(3*Pi))*sqrt(A340794*A340665). +10
1
5, 1, 5, 9, 3, 9, 4, 8, 2, 2, 7, 9, 6, 5, 3, 4, 8, 4, 9, 5, 3, 1, 2, 5, 0, 1, 3, 9, 4, 0, 5, 5, 6, 3, 7, 2, 6, 9, 8, 1, 0, 9, 9, 9, 2, 4, 6, 8, 6, 8, 1, 4, 7, 4, 8, 5, 8, 7, 1, 7, 9, 6, 2, 5, 2, 2, 7, 4, 4, 9, 7, 1, 7, 6, 1, 9, 5, 7, 7, 2, 2, 7, 6, 1, 1, 9, 4, 3, 1, 3, 1, 6, 2, 6, 5, 8, 8, 9, 8, 3, 0, 3, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
For general definition of the constants kappa(n) see Steven Finch 2009 p. 7, for this particular case kappa(-5) see p. 11.
LINKS
Table of n, a(n) for n=0..102.
Steven Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2009 p. 7-11.
FORMULA
Equals exp(-gamma/2)*log((1+sqrt(5))/2)*sqrt(5/Pi)/(2*C(5,2)*C(5,3)), where C(5,2) and C(5,3) are Mertens constants see A340839.
Equals 2*A340866*exp(gamma/4)*((1/5)*log((1+sqrt(5))/2))^(3/4)/sqrt(A340004).
Equals 2*A340866*exp(gamma/4)*log((1+sqrt(5))/2)/(sqrt(5*Pi)*A340884^(1/4)).
Equals 2*A340839*A340866*exp(gamma/2)*log((1+sqrt(5))/2)/sqrt(5*Pi).
Equals sqrt((1/3)*Pi*log(9+4*sqrt(5)))/(sqrt(5^(3/2)*A340004*A340127)). [Finch 2009 p. 11]
EXAMPLE
0.51593948227965348495312501394...
CROSSREFS
Cf. A340004, A340127, A340665, A340794, A340839, A340866.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 26 2021
STATUS
approved
A340884 Decimal expansion of the constant rho(1,5). +10
1
2, 4, 9, 1, 3, 5, 7, 0, 2, 7, 6, 4, 9, 3, 1, 4, 2, 4, 6, 5, 9, 9, 6, 3, 7, 9, 5, 0, 8, 7, 1, 9, 7, 6, 1, 0, 1, 7, 5, 1, 9, 8, 9, 7, 2, 9, 0, 4, 7, 7, 1, 1, 0, 7, 1, 5, 6, 0, 2, 2, 1, 3, 3, 5, 8, 3, 4, 2, 3, 5, 8, 8, 7, 2, 2, 0, 7, 0, 4, 7, 7, 9, 3, 0, 1, 2, 4, 5, 3, 7, 3, 9, 2, 1, 0, 6, 5, 1, 5, 1, 2, 4, 6, 7, 4, 7, 3, 2, 8, 2, 9, 3, 1, 7, 5, 6, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
From definition Steven Finch and Pascal Sebah 2009 p. 1:
rho(n,m) = lim_{s->1} (s-1) Product_{primes p==n (mod m)} (1-1/p^s)^phi(m), where phi(n) = A000010(n) is the Euler totient function.
LINKS
Table of n, a(n) for n=0..116.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 p. 1-2.
FORMULA
Equals 1/(exp(gamma)*A340839^4).
Formulas by Steven Finch and Pascal Sebah 2009 p. 2.
Equals 5*log(2 + sqrt(5))*A340004^2/(3*Pi^2).
Equals 50*log(2 + sqrt(5))*A340808/(13*Pi^2*sqrt(5)*A340628).
EXAMPLE
0.249135702764931424659963795...
CROSSREFS
Cf. A340628, A340808, A340004.
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 25 2021
STATUS
approved
page 1

Search completed in 0.007 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 8 10:21 EDT 2024. Contains 375753 sequences. (Running on oeis4.)