A175640 Decimal expansion of Product_{p = prime} (1 +(3*p^2-1)/((p^2-1)*p*(p+1)) ).
2, 5, 9, 6, 5, 3, 6, 2, 9, 0, 4, 5, 0, 5, 4, 2, 0, 7, 3, 6, 3, 2, 7, 4, 0, 6, 5, 6, 6, 6, 9, 5, 1, 6, 1, 4, 2, 3, 7, 3, 9, 4, 6, 3, 0, 5, 2, 3, 4, 5, 0, 1, 4, 6, 2, 3, 6, 1, 5, 3, 6, 4, 9, 8, 1, 0, 6, 7, 5, 4, 8, 2, 4, 5, 7, 8, 7, 6, 0, 9, 3, 5, 2, 1, 9, 3, 7, 1, 2, 2, 2, 8, 7, 0, 2, 8, 6, 4, 3, 1, 4, 2, 8, 7, 4
Offset: 1
Examples
2.596536290450542073632740...
Links
- M. B. Barban, The large sieve method and its application to number theory, Russ. Math. Surv., Vol. 21, No. 1 (1966), pp. 49-103; MR 0199171.
- Steven R. Finch, Class number theory [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. 88.
- Eric Weisstein's World of Mathematics, Barban's Constant.
- Eric Weisstein's World of Mathematics, Prime Products
- Wikipedia, Euler Product.
Programs
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Maple
read("transforms") : efact := 1+(3*p^2-1)/(p^2-1)/p/(p+1) ; Digits := 130 : tm := 380 : 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 = 50; $MaxExtraPrecision = 5 digits; s = Log[(1 + (3*p^2 - 1)/((p^2 - 1)*p*(p + 1)))] + O[p, Infinity]^(12 digits) // Normal; B = Exp[s /. Power[p, k_] -> PrimeZetaP[-k]]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Jul 24 2017 *)
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PARI
prodeulerrat(1 +(3*p^2-1)/((p^2-1)*p*(p+1))) \\ Amiram Eldar, Mar 18 2021
Formula
Equals (29/18)*(61/48)*(397/360)*(1417/1344)*... inserting p = 2, 3, 5, 7, ... into the factor.
Extensions
More digits from Jean-François Alcover, Jul 24 2017
More digits from Vaclav Kotesovec, Jan 13 2021
Comments