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%I #47 May 29 2023 08:30:08
%S 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,
%T 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,
%U 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
%N Decimal expansion of Product_{p = prime} (1 +(3*p^2-1)/((p^2-1)*p*(p+1)) ).
%C Named Barban's constant after the Soviet mathematician Mark Borisovich Barban (1935-1968). - _Amiram Eldar_, Mar 18 2021
%H M. B. Barban, <a href="http://dx.doi.org/10.1070/RM1966v021n01ABEH004146">The large sieve method and its application to number theory</a>, Russ. Math. Surv., Vol. 21, No. 1 (1966), pp. 49-103; <a href="http://www.ams.org/mathscinet-getitem?mr=199171">MR 0199171</a>.
%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 88.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarbansConstant.html">Barban's Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_product">Euler Product</a>.
%F Equals (29/18)*(61/48)*(397/360)*(1417/1344)*... inserting p = 2, 3, 5, 7, ... into the factor.
%F Equals Sum_{n>=1} A000005(n^2)*A000010(n)/n^3. - _Richard R. Forberg_, May 28 2023
%e 2.596536290450542073632740...
%p 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 :
%p for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i,Le) ; x := evalf(x) ; print(x) ; end do:
%t 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 *)
%o (PARI) prodeulerrat(1 +(3*p^2-1)/((p^2-1)*p*(p+1))) \\ _Amiram Eldar_, Mar 18 2021
%Y Cf. A000005, A000010.
%K cons,nonn
%O 1,1
%A _R. J. Mathar_, Aug 01 2010
%E More digits from _Jean-François Alcover_, Jul 24 2017
%E More digits from _Vaclav Kotesovec_, Jan 13 2021
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