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

1,1

Comments

Named Barban's constant after the Soviet mathematician Mark Borisovich Barban (1935-1968). - Amiram Eldar, Mar 18 2021

Links

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

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.

Formula

Equals (29/18)*(61/48)*(397/360)*(1417/1344)*... inserting p = 2, 3, 5, 7, ... into the factor.

Equals Sum_{n>=1} A000005(n^2)*A000010(n)/n^3. - Richard R. Forberg, May 28 2023

Example

2.596536290450542073632740...

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 *)

Prog

(PARI) prodeulerrat(1 +(3*p^2-1)/((p^2-1)*p*(p+1))) \\ Amiram Eldar, Mar 18 2021

Crossrefs

Cf. A000005, A000010.

Sequence in context: A246206 A200336 A065225 * A204913 A198455 A339174

Adjacent sequences: A175637 A175638 A175639 * A175641 A175642 A175643

Keyword

cons,nonn

Author

R. J. Mathar, Aug 01 2010

Extensions

More digits from Jean-François Alcover, Jul 24 2017

More digits from Vaclav Kotesovec, Jan 13 2021

Status

approved