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A179119
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Decimal expansion of Sum_{p prime} 1/(p*(p+1)).
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19
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3, 3, 0, 2, 2, 9, 9, 2, 6, 2, 6, 4, 2, 0, 3, 2, 4, 1, 0, 1, 5, 0, 9, 4, 5, 8, 8, 0, 8, 6, 7, 4, 4, 7, 6, 0, 6, 4, 4, 2, 5, 9, 4, 1, 9, 4, 7, 4, 0, 7, 0, 4, 5, 6, 1, 5, 0, 2, 2, 8, 6, 0, 0, 7, 6, 2, 4, 2, 2, 1, 6, 6, 7, 9, 2, 9, 0, 7, 9, 4, 4, 3, 2, 1, 7, 0, 3, 2, 0, 7, 5, 1, 3, 2, 3, 5, 1, 0, 3, 1, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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LINKS
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FORMULA
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EXAMPLE
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0.33022992626420324101.. = 1/(2*3) +1/(3*4) +1/(5*6) + 1/(7*8) +... = sum_{n>=1} 1/ (A000040(n)*A008864(n)).
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MAPLE
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interface(quiet=true):
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:
Hurw := proc(a)
local T, p, x, L, i, Le, pre, preT, v, t, M ;
T := 40 ;
preT := 0.0 ;
while true do
1/p/(p+a) ;
subs(p=1/x, %) ;
exp(%) ;
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) ;
M := -a ;
v := 1.0 ;
pre := 0.0 ;
for i from 2 to nops(Le) do
pre := log(v) ;
v := v*evalf(ZetaM(i, M))^op(i, Le) ;
v := evalf(v) ;
end do:
pre := (log(v)+pre)/2. ;
printf("%.105f\n", %) ;
if abs(1.0-preT/pre) < 10^(-Digits/3) then
break;
end if;
preT := pre ;
T := T+10 ;
end do:
pre ;
end proc:
Hurw(1) ;
end proc:
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MATHEMATICA
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digits = 101; S = NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 5]; RealDigits[S, 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)
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PROG
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(PARI) eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
(PARI) sumeulerrat(1/(p*(p+1))) \\ Amiram Eldar, Mar 18 2021
(Magma)
R:=RealField(103);
ExhaustSum :=
function(
k_min, term
: IZ := func<t, k|IsZero(t)>)
c:=R!0; k:=k_min;
repeat
t:=term(k); c+:=t; k+:=1;
until IZ(t, k-1);
return c;
end function;
RealField(101)!
ExhaustSum(2,
func<k|
(-1)^k *
ExhaustSum(1,
func<n|
(mu ne 0 select mu*Log(ZetaFunction(R, k*n))/n else 0)
where mu is MoebiusMu(n)>
: IZ:=func<t, n|MoebiusMu(n)ne 0 and IsZero(t)>
)>);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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