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A179119 Decimal expansion of sum 1/(p(p+1)) over the primes p. 10
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

(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: A010607 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 November 14 07:19 EST 2019. Contains 329111 sequences. (Running on oeis4.)