A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)).

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

Offset

0,1

Links

Jason Kimberley, Table of n, a(n) for n = 0..683

Formula

P(2) - P(3) + P(4) - P(5) + ..., where P is the prime zeta function. - Charles R Greathouse IV, Aug 03 2016

Example

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

Maple

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:
A179119 := proc()
    Hurw(1) ;
end proc:
A179119() ;

Mathematica

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 *)

Prog

(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))))
sumalt(k=2, (-1)^k*primezeta(k)) \\ Charles R Greathouse IV, Aug 03 2016
(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)>
    )>);
// Jason Kimberley, Jan 20 2017

Crossrefs

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

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

Cf. A307379.

Sequence in context: A338116 A325018 A118522 * A098316 A160165 A084055

Adjacent sequences: A179116 A179117 A179118 * A179120 A179121 A179122

Keyword

cons,easy,nonn

Author

R. J. Mathar, Jan 21 2013

Status

approved