Algorithms-ti compute sum of primes,prime(x)

[cino hilliard]

Hi,
I just uploaded to mathforfun some Gcc Pari and basic programs to compute 
the sum of primes in a
range and to compute an approximation of prime(x) to very high accuracy. 
Some of you may be
able to add improvements.

The pari routine primex(n) produces
googol=10^100
primex(googol) 
=23471257358657641780361359099363020719654224259786148259852779144053794033
04282984439399437187551903374.8804209

My conjecture is that this is correct to the first 50 places. Maybe someone 
has software that will
verify this.

So far, I have used sumprimes.exe to compute

n   sum primes <= 10^n
1  17
2  1060
3  76127
4  5736396
5  454396537
6  37550402023
7  3203324994356
8  279209790387276
9  24739512092254535
10  2220822432581729238
11  201467077743744681014
12  18435588552550705911377
13  1699246443377779328889494

I have invented a very good formula to estimate the sum of primes in a 
range. The formula
predicts that  10^14 ~ 157588560672160329790093231. It is in the code.

I am running sumprimes 10000000000000  now and estimate it will take about 
22 days to
compute.  When it is done and if it is in line with my formula, I will 
extend  A046731 with a(12),a
(13),a(14).

The sieve program used is fantastic and very memory friendly. I modified it 
somewhat for this
purpose but could not improve the efficiency at all. The modification is 
good for ranges up to
2^64. Also I used it (modified) to store the 22 bill+ primes < 1 trillion on 
2 hard drives 300 gig+
in about 20 hours. This produces a fast prime(x) function that can be callec 
from Pari.

http://groups.yahoo.com/group/mathforfun/files/Prime%20Number%20Algorithms%20/

Having fun at 2^6,
Cino