A188596 Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).

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

Offset

1,2

Comments

This is the principal scale factor in an estimate of the number of primes p not exceeding N such that p^2+p+1 is also prime [Bateman-Horn].

A102283 in the definition plays the role of the Dirichlet character modulo 3.

After splitting the product into the three modulo-3 classes of primes, this constant turns out to be the product of four factors.

One factor as mentioned by Bateman and Horn is the inverse of A073010.

The second factor is 3/4 arising from the prime 3 which is the sole prime in the class == 0 (mod 3).

The third factor is product_{p == 1 (mod 3)} (1-(3p-1)/(p-1)^3) = 0.8675121817.. which is the constant C(m=3,n=1,s=3) of the arXiv preprint, basically the C(3) variant of A065418 reduced to the modulo class.

The final factor is product_{p == 2 (mod 3)} (1+1/(p^2-1)) = 1/product_{p == 2 (mod 3)} (1-1/p^2) = 1.41406439089214763.. which is the constant zeta(m=3,n=2,s=2) of the preprint and mentioned in A175646.

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.

Links

Table of n, a(n) for n=1..105.

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367, constant C.

H. Davenport and A. Schinzel, A note on certain arithmetical constants, Illinois Math. J. 10 (2) (1966), 181-185.

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Example

Equals 1.5217315350757058188419... = 0.92003856361849186... / A073010 .

Maple

a073010 := evalf(Pi/3/sqrt(3)) ;
Cm3n0s2 := 1-1/(3-1)^2 ;
Cm3n1s3 := 0.867512181712394919089076584762888869720269526863 ;
Zm3n2s2 := 1.4140643908921476375655018190798293799076950693931 ;
Cm3n0s2*Cm3n1s3*Zm3n2s2/a073010 ;

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]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[3^(5/2)*Zs[3, 1, 3]*Z[3, 2, 2]/(4*Pi), digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

Crossrefs

Cf. A053182.

Sequence in context: A212879 A276524 A111395 * A199622 A156730 A159549

Adjacent sequences: A188593 A188594 A188595 * A188597 A188598 A188599

Keyword

nonn,cons,less

Author

R. J. Mathar, Apr 05 2011

Extensions

More terms from Vaclav Kotesovec, Jan 16 2021

Status

approved