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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-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)     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     : IZ:=func     )>); // 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.)