A131671 Decimal expansion of prime analog of the Kepler-Bouwkamp constant: Product_{k>=2} cos(Pi/prime(k)).

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

Offset: 0

Author

R. J. Mathar, Sep 12 2007

Keywords

• cons
• nonn

Examples

			cos(Pi/3)*cos(Pi/5)*cos(Pi/7)*cos(Pi/11)*(...) = 0.312832929508881838333...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 62.
• Adrian R. Kitson, The prime analog of the Kepler-Bouwkamp constant, The Mathematical Gazette, Vol. 92, No. 293 (2008), pp. 293-295, preprint, arXiv:math/0608186 [math.HO], 2006.
• R. J. Mathar, Tightly circumscribed regular polygons, arXiv:1301.6293 [math.MG], 2013.
• Wikipedia, Kepler-Bouwkamp constant

Crossrefs

Cf. A085365.

Programs

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 *)

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

Formula

Product_{p odd prime} cos(Pi/p) where Pi = 3.14159...

The log of this constant is equal to Sum_{k>=1} (1 - 2^(2*k))*zeta(2*k)/k * (P(2*k) - 1/2^(2*k)), where P(s) is the prime zeta function. - Amiram Eldar, Aug 21 2020

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