This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179119 #44 Mar 31 2025 08:06:51
%S A179119 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 A179119 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 A179119 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 A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)).
%H A179119 Jason Kimberley, <a href="/A179119/b179119.txt">Table of n, a(n) for n = 0..683</a>
%H A179119 <a href="/wiki/Index_to_constants#Start_of_section_P">Index to constants which are prime zeta sums</a> {1,0,1}
%F A179119 P(2) - P(3) + P(4) - P(5) + ..., where P is the prime zeta function. - _Charles R Greathouse IV_, Aug 03 2016
%e A179119 0.33022992626420324101.. = 1/(2*3) +1/(3*4) +1/(5*6) + 1/(7*8) +... = sum_{n>=1} 1/ (A000040(n)*A008864(n)).
%p A179119 interface(quiet=true):
%p A179119 read("transforms") ;
%p A179119 Digits := 300 ;
%p A179119 ZetaM := proc(s,M)
%p A179119 local v,p;
%p A179119 v := Zeta(s) ;
%p A179119 p := 2;
%p A179119 while p <= M do
%p A179119 v := v*(1-1/p^s) ;
%p A179119 p := nextprime(p) ;
%p A179119 end do:
%p A179119 v ;
%p A179119 end proc:
%p A179119 Hurw := proc(a)
%p A179119 local T,p,x,L,i,Le,pre,preT,v,t,M ;
%p A179119 T := 40 ;
%p A179119 preT := 0.0 ;
%p A179119 while true do
%p A179119 1/p/(p+a) ;
%p A179119 subs(p=1/x,%) ;
%p A179119 exp(%) ;
%p A179119 t := taylor(%,x=0,T) ;
%p A179119 L := [] ;
%p A179119 for i from 1 to T-1 do
%p A179119 L := [op(L),evalf(coeftayl(t,x=0,i))] ;
%p A179119 end do:
%p A179119 Le := EULERi(L) ;
%p A179119 M := -a ;
%p A179119 v := 1.0 ;
%p A179119 pre := 0.0 ;
%p A179119 for i from 2 to nops(Le) do
%p A179119 pre := log(v) ;
%p A179119 v := v*evalf(ZetaM(i,M))^op(i,Le) ;
%p A179119 v := evalf(v) ;
%p A179119 end do:
%p A179119 pre := (log(v)+pre)/2. ;
%p A179119 printf("%.105f\n",%) ;
%p A179119 if abs(1.0-preT/pre) < 10^(-Digits/3) then
%p A179119 break;
%p A179119 end if;
%p A179119 preT := pre ;
%p A179119 T := T+10 ;
%p A179119 end do:
%p A179119 pre ;
%p A179119 end proc:
%p A179119 A179119 := proc()
%p A179119 Hurw(1) ;
%p A179119 end proc:
%p A179119 A179119() ;
%t A179119 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 A179119 (PARI) eps()=2.>>bitprecision(1.)
%o A179119 primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
%o A179119 sumalt(k=2,(-1)^k*primezeta(k)) \\ _Charles R Greathouse IV_, Aug 03 2016
%o A179119 (PARI) sumeulerrat(1/(p*(p+1))) \\ _Amiram Eldar_, Mar 18 2021
%o A179119 (Magma)
%o A179119 R:=RealField(103);
%o A179119 ExhaustSum :=
%o A179119 function(
%o A179119 k_min, term
%o A179119 : IZ := func<t,k|IsZero(t)>)
%o A179119 c:=R!0; k:=k_min;
%o A179119 repeat
%o A179119 t:=term(k); c+:=t; k+:=1;
%o A179119 until IZ(t,k-1);
%o A179119 return c;
%o A179119 end function;
%o A179119 RealField(101)!
%o A179119 ExhaustSum(2,
%o A179119 func<k|
%o A179119 (-1)^k *
%o A179119 ExhaustSum(1,
%o A179119 func<n|
%o A179119 (mu ne 0 select mu*Log(ZetaFunction(R,k*n))/n else 0)
%o A179119 where mu is MoebiusMu(n)>
%o A179119 : IZ:=func<t,n|MoebiusMu(n)ne 0 and IsZero(t)>
%o A179119 )>);
%o A179119 // _Jason Kimberley_, Jan 20 2017
%Y A179119 Cf. A136141 for 1/(p(p-1)), A085548 for 1/p^2.
%Y A179119 Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
%Y A179119 Cf. A307379.
%K A179119 cons,easy,nonn
%O A179119 0,1
%A A179119 _R. J. Mathar_, Jan 21 2013