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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
OFFSET
0,1
LINKS
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.
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
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: A052914 A332491 A144189 * A060750 A204025 A204126
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