PrimePi(10^23): A006880 and MathWorld disagreement

[Richard Guy]

While you're at it, I'd be happy if someone
could check & complete the following table
from the introduction to Chapter A of UPINT.
Many thanks in anticipation.    R.

PS I just asked for A006880 and it only went
to 10^{20}.   R.

\begin{center}
\begin{tabular}{lrrrr}
$x$ & $\pi(x)$ & $(x/\ln x)-\pi(x)$ & Li$(x)-\pi(x)$ & $R(x)-\pi(x)$ \\
  10    & 4        &   0        & 2    & \\
$10^2$ & 25       & $-3$       & 5    & 1 \\
$10^3$ & 168      & $-23$      & 10   & 0 \\
$10^4$ & 1229     & $-143$     & 17   & $-2$ \\
$10^5$ & 9592     & $-906$     & 38   & $-5$ \\
$10^6$ & 78498    & $-6116$    & 130  & 29 \\
$10^7$ & 664579   & $-44158$   & 339  & 88 \\
$10^8$ & 5761455  & $-332774$  & 754  & 97 \\
$10^9$ & 50847534 & $-2592592$ & 1701 & $-79$ \\
$10^{10}$ & 455052511 & $-20758029$
  & 3104      & $-1828$ \\
$10^{11}$ & 4118054813 & $-169923159$
  & 11588     & $-2318$ \\
$10^{12}$ & 37607912018 & $-1416705193$
  & 38263     & $-1476$ \\
$10^{13}$ & 346065536839 & $-11992858452$
  & 108971    & $-5773$ \\
$10^{14}$ & 3204941750802 & $-102838308636$
  & 314890    & $-19200$ \\
$10^{15}$ & 29844570422669 & $-891604962452$
  & 1052619   & 73218 \\
$10^{16}$ & 279238341033925 & $-7804289844393$
  & 3214632   & 327052 \\
$10^{17}$ & 2623557157654233 & $-68883734693928$
  & 7956589   & $-598255$ \\
$10^{18}$ & 24739954287740860 & $-612483070893536$
  & 21949555  & $-3501366$ \\
$10^{19}$ & 234057667276344607 & $-5481624169369961$
  & 99877775  & 23884332 \\
$10^{20}$ & 2220819602560918840 & $-49347193044659702$
  & 223744644 & $-4891826$ \\
$10^{21}$ & 21127269486018731928 & $-446579871578168707$
  & 597394253 & $-86432205$ \\
$10^{22}$ & 201467286689315906290 & $-$
  &  & $-$ \\
$10^{23}$ & 1925320391606803968923 & $-$
  &  & $-$ \\
\end{tabular}
\end{center}

On Tue, 26 Feb 2008, Max Alekseyev wrote:

> Neil, Eric, SeqFans,
>
> The values for pi(10^23) listed at
> http://www.research.att.com/~njas/sequences/b006880.txt and at
> http://mathworld.wolfram.com/PrimeCountingFunction.html differ
> dramatically. Moreover, MathWorld claims that the exact value of
> pi(10^23) is known only up to +/- 1.
>
> So, I wonder which value is correct and what is the current status of
> pi(10^23) ?
>
> Regards,
> Max
>
>
>




This looks like A088919 ....

zs> From seqfan-owner at ext.jussieu.fr  Tue Feb 26 20:21:48 2008
zs> Date: Tue, 26 Feb 2008 11:21:17 -0800 (PST)
zs> From: zak seidov <zakseidov at yahoo.com>
zs> Subject: m=p^2+q^2, (p<q primes): number of presentations
zs> To: seqfan at ext.jussieu.fr
zs> 
zs> Dear seqfans,
zs> 
zs> first of all,
zs> many thanks to you those responding 
zs> (and sorry to much more those annoying..)
zs> to my question_posts.
zs> 
zs> Here is my last one.
zs> Can anyone plz check and extend this?
zs> thanks, zak
zs> 
zs> 
zs> A1:
zs> Smallest positive number expressible as sum of squares
zs> of two distinct primes p<q exactly in n ways:
zs> 1, 13, 410, 2210, 10370, 202130, 229970, 197210, 81770
zs> 0 way: 1
zs> 1 way: 13=2^2+3^2,
zs> 2 ways: 410=7^2+19^2=11^2+17^2,
zs> 3 ways: 2210=19^2+43^2=23^2+41^2=29^2+37^2,
zs> 4 ways:
zs> 10370=13^2+101^2=31^2+97^2=59^2+83^2=71^2+73^2,
zs> 5 ways:
zs> 202130=23^2+449^2=97^2+439^2=163^2+419^2=211^2+397^2=251^2+373^2,
zs> 
zs> 6 ways:
zs> 229970=23^2+479^2=109^2+467^2=193^2+439^2=263^2+401^2=269^2+397^2=331^2+347^2,
zs> 7 ways:
zs> 197210=31^2+443^2=67^2+439^2=107^2+431^2=173^2+409^2=199^2+397^2=241^2+373^2=311^2+317^2,
zs> 8 ways:
zs> 81770=41^2+283^2=53^2+281^2=71^2+277^2=97^2+269^2=137^2+251^2=157^2+239^2=179^2+223^2=193^2+211^2,
zs> 9 ways: ? 
zs> 10 ways: ?
zs> ...........
zs> ...........
zs> 
zs> BTW for those interested: 
zs> for m's up to 10^6,
zs> there are only 8732 m's 
zs> expressible as p^2+q^2 (p<q primes)
zs> at least in 1 way,
zs> right? 



Max Alekseyev:

> The values for pi(10^23) listed at
> http://www.research.att.com/~njas/sequences/b006880.txt and at
> http://mathworld.wolfram.com/PrimeCountingFunction.html differ
> dramatically. Moreover, MathWorld claims that the exact value of
> pi(10^23) is known only up to +/- 1.

My guess is that Eric's value came from an old (2002) attempt (see http://numbers.computation.free.fr/Constants/constants.html) 
The OEIS 10^23 value is Tomás Oliveira e Silva's (see http://www.ieeta.pt/~tos/primes.html) 
.

I'm not sure Eric puts a lot of effort into updating MathWorld  
anymore. I emailed him one month ago about his incorrect length-of- 
Gregorian-Easter cycle (see http://scienceworld.wolfram.com/astronomy/Easter.html)