tables of p(n) and/or pi(n) / tables of primes

[cino hilliard]

Hi Hugo,


>From: Hugo Pfoertner <all at abouthugo.de>
>To: seqfan at ext.jussieu.fr
>Subject: Re: tables of p(n) and/or pi(n) / tables of primes
>Date: Wed, 21 Dec 2005 01:57:59 +0100
>Why should CPU time change from being proportional to the number of
>items processed? If I had used the necessary amount of memory I would
>expect ~= 450s.
Yes I made a mistake adding an extra 0.
>
> > num 10000000000
> > 1234379338586942892505.0000000000000000000
> > Time  2042.953s  (While multi-tasking. Should be 1800 straight through)
> > Hugo fortran time ?  45000s ?
Also SB ~ 4500
>
>
>Since disk space on my employer's computers is an expensive resource (in
>contrast to CPU time), I will not participate in any effort to extend
>anything needing big amounts of stored data on disk.

Interesting. When I was in the corporate world I often got the warning 
letter from the IS manager
stating I was the second of first biggest user of CPU during prime time. 
Little did he know that I
was using 3 ID's. That was in the  nineties when 100 megs was a a big hard 
drive on a PC. Now
you can buy a terabyte of 4 250 gig HD's for a total of $520.00 at 
Microcenter.

Update on this project
------------------------------------------------
So far we have
x  y = sum of first 10^x primes
0,2
1,129
2,24133
3,3682913
4,496165411
5,62260698721
6,7472966967499
7,870530414842019
8,99262851056183695
9,11138479445180240497
10,1234379338586942892505

4 of these sums are prime 2,24133,870530414842019,11138479445180240497
the other 7 are composite. Looke like a couiple more sequences that will 
stay under 10 terms
for years to come.

Notice the progression of the first digits. If we take log10(y) we get
x    log10(y)
1,2.1105897103
2,4.3826113129
3,6.5661914598
4,8.6956264852
5,10.7942139902
6,12.8734930626
7,14.9397839492
8,16.9967867450
9,19.0468259077
10,21.0914486438

These numbers are extremely an smooth polynomial  fit  as curvefitting a 4th 
degree poloynomial
through the 5 even values of x will show.

Output from my program sicurvqf
-------------------------------------------------
FOR J=1 TO 1
SOLUTION VECTOR S( 1)
X 0 =-0.200762742700000000000000000000000000000000000000000000000000000
X 1 =2.3886411202541666666666666666666666666666666666666666666666666666
X 2 =-0.056956359728125000000000000000000000000000000000000000000000000
X 3 =0.0045242118989583333333333333333333333333333333333333333333333333
X 4 =-0.000142277574218750000000000000000000000000000000000000000000000

CURVEFITTING REGRESSION STATISTICS
--------------------------------------------------------
DEG POLY  =  4
NO. OBS   =  5
AVG X     = 6.00000000000000000000
AVG Y     = 12.8079932499000000000
STD ERROR = 2.33990250030424564520e-105
STD DEV   = 5.90024636137376606234
COF CORR  = 1.00000000000000000000
COF DETR  = 1.00000000000000000000

We then evaluate the odd values of x with this 4th degree exact fit of even 
values of x to get
x          y(x)
3,6.56318261826953125
5,10.79513686885078125
7,14.93905969804453125
9,19.04820951150078125
These are very close to the actuals above.
Next we curvefit the 6 degree poly through the 7 x values 4,5,6,7,8,9,10 and 
extrapolate y(11)
x  y(x)
11,23.1313771663
We then adjust this y value and a 4th degree through the 5 points for x = 
3,5,7,11 and evaluate
y(10). We repeat this adjustment until the interpolation is as close as 
expected to the true
value of y for x=10. This y for x=11 is 23.1374000000. Then 10^23.1374000000 
=
137214497657306000000000 = sum of the first 100 billion primes. Taking this 
process one more
step using y(11) = 10^23.1374000000 as gospel, we arrive at y(12) = 
25.1880000000
and 10^25.1880000000 = 15,417,004,529,495,600,000,000,000 = my estimate of 
the sum of the
first 1 trillion primes. So if each prime is a dollar value, we are talking 
BIG money here - 15.4
septillion dollars.

If we use pounds weight we have 15,417 sextillion lbs or 7.7 sextillion tons 
- pretty close to the
weight of the earth - 6 sextillion tons. Now of course it may be closer 
since my 15.7 septillion
is a 2 step extrapolation. Will we ever know?

Have fun,

Cino