A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

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

Offset

0,1

Comments

Equals (49/64)*(668/729)*(15304/15625)*(116724/117649)*... inserting p= A000040 = 2, 3, 5, 7.. into the factor. Slightly larger than Product_{p=primes} (1-3/p^3) = 0.534566872085103888416775...

Links

Table of n, a(n) for n=0..104.

Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 85.

Takashi Taniguchi, A mean value theorem for the square of class number times regulator of quadratic extensions, Annales de l'institut Fourier. Vol. 58, No. 2 (2008), pp. 625-670; arXiv preprint, arXiv:math/0410531 [math.NT], 2004-2006.

Eric Weisstein's World of Mathematics, Prime Products.

Eric Weisstein's World of Mathematics, Taniguchi's Constant.

Wikipedia, Euler Product.

Example

0.678234491917391978035...

Maple

read("transforms") : efact := 1-3/p^3+2/p^4+1/p^5-1/p^6 ; Digits := 130 : tm := 310 : subs (p=1/x, 1/efact) ; taylor(%, x=0, tm) : L := [seq(coeftayl(%, x=0, i), i=1..tm-1)] : Le := EULERi(L) : x := 1.0 :
for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i, Le) ; x := evalf(x) ; print(x) ; end do:

Mathematica

digits = 105; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s];
LR = LinearRecurrence[{0, 0, 3, -2, -1, 1}, {0, 0, -9, 8, 5, -33}, 2 m0];
r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

Prog

(PARI) prodeulerrat(1-3/p^3+2/p^4+1/p^5-1/p^6) \\ Amiram Eldar, Mar 04 2021

Crossrefs

Cf. A256392, A330523.

Sequence in context: A004487 A274137 A010500 * A256922 A202346 A117022

Adjacent sequences: A175636 A175637 A175638 * A175640 A175641 A175642

Keyword

cons,nonn

Author

R. J. Mathar, Aug 01 2010

Extensions

More digits from Jean-François Alcover, Apr 15 2016

Status

approved