sum of 1/A007504(n)

[Jack Brennen]

Don Reble wrote:
> Seqfans:
> 
>      I might as well do this bit.
> 
>      Summing n=4016708412 primes, I get p(n)=97434417233,
>      primeSum=191462469311735988657,
>      seriesSum=1.02347632390000000000618+.
>      Anyone want to double-check?
> 
>      And I compute an upper bound of 1.02347632395-.
> 

Here's a chart I put together.

First column is the limit A.  Each row increases A by 20% over the previous
   row.

Second column is the sum of 1/A007504(n) summed for primes <= A, plus the
   integral from log(A) to +infinity of -exp(x)/(x*eint1(-2*x)) dx.  (The sum
   is asymptotic to this integral.)

So basically, the second column is the sum for primes <= A, plus a close
approximation to the remaining infinite sum.

You can see that the second column converges to 1.02347632392012...
Furthermore, the approximation is already correct to 10 significant
digits for A=1200000 and stays so from there for the rest of the table.
The table also shows that the approximation converges quickly and has
no significant upward or downward trend, a very good indication that
the approximation is accurate; as we replace each successive range's
integral with the actual computed sum, the numbers converge on the
target much faster than the finite sum does.

I've added a final row based on your reported result above, which shows
the convergence continues...


1000000 1.023476322905411393512669156
1200000 1.023476324471905303298786875
1440000 1.023476323856524999388118998
1728000 1.023476323710202588497736563
2073600 1.023476324002968802427436530
2488320 1.023476323770113088901754079
2985984 1.023476323885391431851061286
3583180 1.023476323736402017832985868
4299816 1.023476323837815372498973702
5159779 1.023476323939889074028470272
6191734 1.023476323872271968332158850
7430080 1.023476323921513829936208531
8916096 1.023476323921622275493714698
10699315 1.023476323915756450812186687
12839178 1.023476323937222540400139147
15407013 1.023476323902478619598740452
18488415 1.023476323918419195073069064
22186098 1.023476323920772952679401716
26623317 1.023476323918059440534341229
31947980 1.023476323919883307735703909
38337576 1.023476323918282402904990889
46005091 1.023476323921603062017399838
55206109 1.023476323918582541790001298
66247330 1.023476323919524838740432720
79496796 1.023476323919426575901536786
95396155 1.023476323919696107532808387
114475386 1.023476323920053360473529071
137370463 1.023476323919882261872585524
164844555 1.023476323919917669423875543
197813466 1.023476323919910385878861263
237376159 1.023476323920004439606228313
284851390 1.023476323920108996779577507
341821668 1.023476323920073821889912627
410186001 1.023476323920090514959203546
492223201 1.023476323920152612885961732
590667841 1.023476323920100891829620207
708801409 1.023476323920144084086399388
850561690 1.023476323920135767580694923
1020674028 1.023476323920136802804333713
1224808833 1.023476323920118552499123994
1469770599 1.023476323920113168185970333
1763724718 1.023476323920125662069507636
2116469661 1.023476323920131979245378790
2539763593 1.023476323920127405230667093
3047716311 1.023476323920125103175012961
3657259573 1.023476323920129071816347204
4388711487 1.023476323920128247487229319
5266453784 1.023476323920127297308729250
6319744540 1.023476323920129137753256200
7583693448 1.023476323920126758123702116
9100432137 1.023476323920127854086792932
10920518564 1.023476323920127697634804584
13104622276 1.023476323920127879518612012
15725546731 1.023476323920128315044374395
18870656077 1.023476323920127547044212691
22644787292 1.023476323920128124289292355
27173744750 1.023476323920127940714665343
32608493700 1.023476323920127986323204372

97434417233 1.023476323920127983534057193  (from your result above)