A179119 - OEIS

%I #42 Sep 08 2022 08:45:54

%S 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,

%T 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,

%U 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

%N Decimal expansion of Sum_{p prime} 1/(p*(p+1)).

%H Jason Kimberley, <a href="/A179119/b179119.txt">Table of n, a(n) for n = 0..683</a>

%F P(2) - P(3) + P(4) - P(5) + ..., where P is the prime zeta function. - _Charles R Greathouse IV_, Aug 03 2016

%e 0.33022992626420324101.. = 1/(2*3) +1/(3*4) +1/(5*6) + 1/(7*8) +... = sum_{n>=1} 1/ (A000040(n)*A008864(n)).

%p interface(quiet=true):

%p read("transforms") ;

%p Digits := 300 ;

%p ZetaM := proc(s,M)

%p     local v,p;

%p     v := Zeta(s) ;

%p     p := 2;

%p     while p <= M do

%p         v := v*(1-1/p^s) ;

%p         p := nextprime(p) ;

%p     end do:

%p     v ;

%p end proc:

%p Hurw := proc(a)

%p         local T,p,x,L,i,Le,pre,preT,v,t,M ;

%p     T := 40 ;

%p     preT := 0.0 ;

%p     while true do

%p             1/p/(p+a) ;

%p             subs(p=1/x,%) ;

%p             exp(%) ;

%p             t := taylor(%,x=0,T) ;

%p             L := [] ;

%p             for i from 1 to T-1 do

%p                     L := [op(L),evalf(coeftayl(t,x=0,i))] ;

%p             end do:

%p             Le := EULERi(L) ;

%p         M := -a ;

%p             v := 1.0 ;

%p             pre := 0.0 ;

%p             for i from 2 to nops(Le) do

%p                     pre := log(v) ;

%p                     v := v*evalf(ZetaM(i,M))^op(i,Le) ;

%p                     v := evalf(v) ;

%p             end do:

%p         pre := (log(v)+pre)/2. ;

%p         printf("%.105f\n",%) ;

%p         if abs(1.0-preT/pre)  < 10^(-Digits/3) then

%p             break;

%p         end if;

%p         preT := pre ;

%p         T := T+10 ;

%p     end do:

%p         pre ;

%p end proc:

%p A179119 := proc()

%p     Hurw(1) ;

%p end proc:

%p A179119() ;

%t 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 *)

%o (PARI) eps()=2.>>bitprecision(1.)

%o primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))

%o sumalt(k=2,(-1)^k*primezeta(k)) \\ _Charles R Greathouse IV_, Aug 03 2016

%o (PARI) sumeulerrat(1/(p*(p+1))) \\ _Amiram Eldar_, Mar 18 2021

%o (Magma)

%o R:=RealField(103);

%o ExhaustSum :=

%o   function(

%o     k_min, term

%o   : IZ := func<t,k|IsZero(t)>)

%o     c:=R!0; k:=k_min;

%o     repeat

%o       t:=term(k); c+:=t; k+:=1;

%o     until IZ(t,k-1);

%o     return c;

%o   end function;

%o RealField(101)!

%o ExhaustSum(2,

%o   func<k|

%o     (-1)^k *

%o     ExhaustSum(1,

%o       func<n|

%o         (mu ne 0 select mu*Log(ZetaFunction(R,k*n))/n else 0)

%o         where mu is MoebiusMu(n)>

%o     : IZ:=func<t,n|MoebiusMu(n)ne 0 and IsZero(t)>

%o     )>);

%o // _Jason Kimberley_, Jan 20 2017

%Y Cf. A136141 for 1/(p(p-1)), A085548 for 1/p^2.

%Y Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).

%Y Cf. A307379.

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Jan 21 2013