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A131671 Decimal expansion of prime analog of the Kepler-Bouwkamp constant: Product_{k>=2} cos(Pi/prime(k)). 2
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..80.

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

PROG

(PARI) (zp(n)=suminf(k=1, moebius(k)*log(zeta(n*k))/k)); exp(-suminf(k=1, (4^k-1)*zeta(2*k)/k*(zp(2*k)-1/4^k))) \\ M. F. Hasler, May 18 2014

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

STATUS

approved

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Last modified December 21 15:47 EST 2014. Contains 252324 sequences.