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A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)). 11
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)
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-Fran├žois 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

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Last modified April 13 19:10 EDT 2021. Contains 342939 sequences. (Running on oeis4.)