sum of 1/A007504(n)

[Max Alekseyev]

The sum over primes below 10^10 equals:

1.023476323724390...

Max

On 5/14/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> To get another view of the picture and avoid rounding problems,
> I add the terms (all primes up to 10^8 now) to -(1/2+Pi/6).
> The final error is -0.000122471148057352456133159527667
> note that it doesn't change much using the primes between 10^6 and 10^8.
> M.H.
>
> (16:20) gp > default(primelimit,10^8)
> time = 529 ms.
> %352 = 100000000
> (16:25) gp > A122989()
> Using primes up to p=7 we get s-1/2-Pi/6=-0.164775246186534167194754289370
> Using primes up to p=29 we get s-1/2-Pi/6=-0.0566903862879554898885944397401
> Using primes up to p=97 we get s-1/2-Pi/6=-0.0185414488842308031600386407150
> Using primes up to p=337 we get s-1/2-Pi/6=-0.00558518994551079864684524203284
> Using primes up to p=1153 we get s-1/2-Pi/6=-0.00173111308111007972061057843608
> Using primes up to p=3943 we get s-1/2-Pi/6=-0.000598164056509723507742308375967
> Using primes up to p=13441 we get
> s-1/2-Pi/6=-0.000263768074987820108596928175528
> Using primes up to p=45197 we get
> s-1/2-Pi/6=-0.000164695686510146697438771063219
> Using primes up to p=151091 we get
> s-1/2-Pi/6=-0.000135149315980808074615916216534
> Using primes up to p=502259 we get
> s-1/2-Pi/6=-0.000126286221426781890307092679504
> Using primes up to p=1662377 we get
> s-1/2-Pi/6=-0.000123614264028601910280473870246
> Using primes up to p=5477083 we get
> s-1/2-Pi/6=-0.000122805344033941399314757969523
> Using primes up to p=17993477 we get
> s-1/2-Pi/6=-0.000122559574822490262627692674695
> Using primes up to p=58954411 we get
> s-1/2-Pi/6=-0.000122484676502745846329925922711
> final sum of primes:279209790387276
> time = 19,418 ms.
> %357 = -0.000122471148057352456133159527667
>
> modified pari code:
> A122989()={ local( preci=10, sp=0, s=-(1/2+Pi/6));
> forprime(p=1,default(primelimit), sp+=p; s+=1/sp;
>  if(sp>preci, print("Using primes up to p=",p," we get
> s-1/2-Pi/6=",s);preci*=10);
> ); print("final sum of primes:",sp);
> s}
>
>
> > On 5/14/07, N. J. A. Sloane <njas at research.att.com> wrote:
> > >
> > > Dear Seqfans,  A correspondent,
> > > "fabio mercurio" <mercurio.fabio at gmail.com>
> > > writes to say that this number
> > >
> > > %I A122989
> > > %S A122989 1,0,2,3,4,7,6,3,2
> > > %N A122989 Decimal expansion of Sum_{n >= 1} 1/A007504(n), where A007504(n) is the sum of the f\
> > > irst n primes.
> > > %e A122989 1/2+1/5+1/10+1/17+1/28+1/41+1/58+1/77+1/100+... = 1.02347632...
> > > %Y A122989 Cf. A007504.
> > > %Y A122989 Adjacent sequences: A122986 A122987 A122988 this_sequence A122990 A122991 A122992
> > > %Y A122989 Sequence in context: A026237 A125150 A072275 this_sequence A077223 A055265 A117922
> > > %K A122989 cons,nonn,more
> > > %O A122989 1,3
> > > %A A122989 Pierre CAMI (pierrecami(AT)tele2.fr), Oct 28 2006
> > >
> > > is really equal to 1/2 + Pi/6.  But the agreement is
> > > not very close.  Can someone compute more decimal places?
> > >
> > > Thanks
> > >
> > > Neil
> > >
> >
>