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 A175640 Decimal expansion of Product_{p = prime} (1 +(3*p^2-1)/((p^2-1)*p*(p+1)) ). 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 (list; constant; graph; refs; listen; history; text; internal format)
 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

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Last modified February 22 00:43 EST 2024. Contains 370239 sequences. (Running on oeis4.)