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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
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
Sequence in context: A246206 A200336 A065225 * A204913 A198455 A339174
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.)