This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A179119 Decimal expansion of sum 1/(p(p+1)) over the primes p. 10

%I

%S 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 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 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 Decimal expansion of sum 1/(p(p+1)) over the primes p.

%H Jason Kimberley, <a href="/A179119/b179119.txt">Table of n, a(n) for n = 0..683</a>

%F P(2) - P(3) + P(4) - P(5) + ..., where P is the prime zeta function. - _Charles R Greathouse IV_, Aug 03 2016

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

%p interface(quiet=true):

%p Digits := 300 ;

%p ZetaM := proc(s,M)

%p local v,p;

%p v := Zeta(s) ;

%p p := 2;

%p while p <= M do

%p v := v*(1-1/p^s) ;

%p p := nextprime(p) ;

%p end do:

%p v ;

%p end proc:

%p Hurw := proc(a)

%p local T,p,x,L,i,Le,pre,preT,v,t,M ;

%p T := 40 ;

%p preT := 0.0 ;

%p while true do

%p 1/p/(p+a) ;

%p subs(p=1/x,%) ;

%p exp(%) ;

%p t := taylor(%,x=0,T) ;

%p L := [] ;

%p for i from 1 to T-1 do

%p L := [op(L),evalf(coeftayl(t,x=0,i))] ;

%p end do:

%p Le := EULERi(L) ;

%p M := -a ;

%p v := 1.0 ;

%p pre := 0.0 ;

%p for i from 2 to nops(Le) do

%p pre := log(v) ;

%p v := v*evalf(ZetaM(i,M))^op(i,Le) ;

%p v := evalf(v) ;

%p end do:

%p pre := (log(v)+pre)/2. ;

%p printf("%.105f\n",%) ;

%p if abs(1.0-preT/pre) < 10^(-Digits/3) then

%p break;

%p end if;

%p preT := pre ;

%p T := T+10 ;

%p end do:

%p pre ;

%p end proc:

%p A179119 := proc()

%p Hurw(1) ;

%p end proc:

%p A179119() ;

%t 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 (PARI) eps()=2.>>bitprecision(1.)

%o primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))

%o sumalt(k=2,(-1)^k*primezeta(k)) \\ _Charles R Greathouse IV_, Aug 03 2016

%o (MAGMA)

%o R:=RealField(103);

%o ExhaustSum :=

%o function(

%o k_min, term

%o : IZ := func<t,k|IsZero(t)>)

%o c:=R!0; k:=k_min;

%o repeat

%o t:=term(k); c+:=t; k+:=1;

%o until IZ(t,k-1);

%o return c;

%o end function;

%o RealField(101)!

%o ExhaustSum(2,

%o func<k|

%o (-1)^k *

%o ExhaustSum(1,

%o func<n|

%o (mu ne 0 select mu*Log(ZetaFunction(R,k*n))/n else 0)

%o where mu is MoebiusMu(n)>

%o : IZ:=func<t,n|MoebiusMu(n)ne 0 and IsZero(t)>

%o )>);

%o // _Jason Kimberley_, Jan 20 2017

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

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

%Y Cf. A307379.

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Jan 21 2013

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)