# sum of 1/A007504(n)

Jack Brennen jb at brennen.net
Wed May 16 07:31:16 CEST 2007

```As I posted yesterday (in part)...

> And the integral appears to converge (as N -> infinity) to:
>
>    1.0234763239201...

I'm fairly sure all those digits are correct.  I've computed the
finite integrals and the sum out to the point where they are
tracking one another consistently within about 10^-14.
For instance, the sum in question computed over the primes from
2^31 to 2^32 is:

0.0000000004547153708405386984

And the integral over the same range computes to:

0.0000000004547134182096031795

Brendan McKay wrote:
> Assuming that the magnitude of the tail of the sum is something of
> order 1/(n*log(n)) after n terms, I guess the final value is about
> 1.0234763237 or 1.0234763238. However, this type of convergence
> acceleration of slowly converging sums has a habit of producing
> illusions, so this guess may well be way off.  It would be
> better to make a theoretical estimate of the tail and add that
> to the partial sums.
>
> Brendan.
>
>
> * Simon Plouffe <simon.plouffe at gmail.com> [070516 07:44]:
>> Hello, here are the latest results.
>>
>> 1.0234763205449071151695629254011 after reaching
>> the 30244000'th prime.
>>
>> Here is a trace of the last computation
>>
>>  Sum                                  term             index
>> =============================================================
>> 1.0234763205433771485177329655746, 8493160443506957, 30231000
>> 1.0234763205434948863081814357548, 8493738372115593, 30232000
>> 1.0234763205436126160876758199314, 8494316321695723, 30233000
>> 1.0234763205437303378570190511747, 8494894291396737, 30234000
>> 1.0234763205438480516170236077605, 8495472281427875, 30235000
>> 1.0234763205439657573684938130720, 8496050292177521, 30236000
>> 1.0234763205440834551122280724106, 8496628324121853, 30237000
>> 1.0234763205442011448490288997553, 8497206376389379, 30238000
>> 1.0234763205443188265797046941143, 8497784448902379, 30239000
>> 1.0234763205444365003050644095177, 8498362541614343, 30240000
>> 1.0234763205445541660259204510602, 8498940654066589, 30241000
>> 1.0234763205446718237430861024454, 8499518786938911, 30242000
>> 1.0234763205447894734573634207349, 8500096940380361, 30243000
>> 1.0234763205449071151695629254011, 8500675113418419, 30244000
>>
>> again : close to 1/2+Pi/6 but considering that the last
>> term is of the order of 1/8500675113418419 and the
>> diff. with the supposed value is ony 1/8166 approx.
>>
>> I don't see where all this would converge to it.
>>
>> Simon Plouffe
>>
>>
>>
>
>

```