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 A131671 Decimal expansion of prime analog of the Kepler-Bouwkamp constant: Product_{k>=2} cos(Pi/prime(k)). 3
 3, 1, 2, 8, 3, 2, 9, 2, 9, 5, 0, 8, 8, 8, 1, 8, 3, 8, 3, 3, 3, 2, 5, 9, 3, 6, 3, 9, 6, 8, 5, 3, 6, 4, 2, 1, 7, 5, 6, 8, 3, 3, 6, 8, 7, 7, 6, 7, 1, 1, 7, 3, 8, 5, 3, 1, 9, 8, 6, 5, 1, 3, 0, 1, 9, 7, 6, 7, 9, 7, 2, 6, 1, 9, 0, 7, 0, 3, 4, 8, 1, 3, 0, 7, 6, 2, 3, 3, 2, 2, 3, 0, 0, 0, 7, 6, 8, 4, 5, 5, 0, 5, 1, 2, 7, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 A. R. Kitson, The prime analog of the Kepler-Bouwkamp constant, arXiv:math/0608186 and The Mathematical Gazette 92: 293. R. J. Mathar, Tightly circumscribed regular polygons, arXiv:1301.6293 [math.MG] Wikipedia, Kepler-Bouwkamp constant FORMULA product{odd primes p} cos(Pi/p) where Pi=3.14159... EXAMPLE cos(Pi/3)*cos(Pi/5)*cos(Pi/7)*cos(Pi/11)*(...) = 0.312832929508881838333... MAPLE read("transforms") ; Digits := 300 ; ZetaM := proc(s, M)     local v, p;     v := Zeta(s) ;     p := 2;     while p <= M do         v := v*(1-1/p^s) ;         p := nextprime(p) ;     end do:     v ; end proc: T := 40 ; preT := 0.0 ; while true do     cos(Pi/p) ;     subs(p=1/x, %) ;     t := taylor(%, x=0, T) ;     L := [] ;     for i from 1 to T-1 do         L := [op(L), evalf(coeftayl(t, x=0, i))] ;     end do:     Le := EULERi(L) ;     v := 1.0 ;     pre := 0.0 ;     for i from 2 to nops(Le) do         pre := v ;         v := v*evalf(ZetaM(i, 2))^op(i, Le) ;     end do:     pre := (v+pre)/2. ;     printf("%.80f\n", pre) ;     if abs(1.0-preT/pre)  < 10^(-Digits/3) then         break;     end if;     preT := pre ;     T := T+15 ; end do: # R. J. Mathar, Jan 23 2013 MATHEMATICA Block[{\$MaxExtraPrecision=1000}, Do[Print[Exp[-Sum[N[(2^(2k)-1)*Zeta[2k]/k*(PrimeZetaP[2k]-1/2^(2k)), 120], {k, 1, m}]]], {m, 300, 350}]] (* Vaclav Kotesovec, Jun 02 2015 *) PROG (PARI) primezeta(n)=sum(k=1, lambertw(10.^default(realprecision)*log(4)) \log(4)+1, moebius(k)*log(zeta(n*k))/k) exp(-suminf(k=1, (4^k-1)*zeta(2*k)/k*(primezeta(2*k)-1/4^k))) \\ M. F. Hasler and Charles R Greathouse IV, May 28 2015 CROSSREFS Cf. A085365. Sequence in context: A084602 A100888 A052914 * A060750 A204025 A204126 Adjacent sequences:  A131668 A131669 A131670 * A131672 A131673 A131674 KEYWORD cons,nonn AUTHOR R. J. Mathar, Sep 12 2007 EXTENSIONS More digits from R. J. Mathar, Mar 01 2009, Jan 23 2013 Edited by M. F. Hasler, May 18 2014 More digits from Vaclav Kotesovec, Jun 02 2015 STATUS approved

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