Weight versus number of primes
 

I sieved each K value up to k=36000 or so and recorded the # of candidates
left after a sieve to p=100M versus the number of primes in n=0-10k. The
resulting graph is kinda neat :geek: so I though I'd share it.
The zipped Excel spreadsheet with a chart of weight vs. # of primes is
about 500k and can be found at
[URL="www.15k.org/var/WeightVsPrimes_xls.zip"]WeightVsPrimes_xls.zip[/URL]

The spreadsheet is sorted by # of primes/weight so a lot of the very
low-weight K values with a few primes show up near the top though
they're not necessarily the best ones for searching (depending on what
you're looking for I guess).

Also note, that many of the low-valued Ks have been searched so it's
a good idea to check the top-5000 list and low-weight RPS thread
if something catches your eye.

Just what I needed-- more data to stare at and convince myself I'm analysing in a useful way.

This is great stuff, though, and might really help in finding new smallish candidates to test. Perhaps we pick a few that meet certain criteria, test to n=100,000, and see if the pattern holds. We'd have to exclude low-weight candidates-- perhaps 1000 candidates left in sieve as minimum? Perhaps we filter for min weight, then test the top 10 or 20 (or more?) candidates to 100k, and repeat the analysis.
-Curtis

Larry, thank you for sharing this really interesting analysis. Great wealth of data to play with.

Edo

I think I'll take k=1515, however, I have tested it till n=20000 and have found 39 instead of 29 primes for n<10000...
I have also checked k=12345, which I am working on for some time now, and for n<10000 there are 15 instead of 13 primes...

I think some primes get sieved out (something like when 2^n is less than k)
so exact prime counts may differ.

If you sieved by NewPGen to 100M before testing primality then it removed all primes smaller than 100M. For example, 1515*2^n-1 is prime for n=3, 4, 5, 6, 8, and 12 but they would be all removed by NewPGen (1515*2^12-1 = 6205439). The next prime at n=22 is larger than 6bn and shuld have survived sieving to 100M. If you want I can create a script in Pari (or pfgw) to recover small primes. But in the k=1515 example above small primes account only for 6 out of 10 missing primes mentioned by Cruelty. Where are the other 4?

The number of primes is really the number up to 5K and NOT 10k. Also,
it's 1 more than the number of primes since I did a unix 'wc' of the prime
file and that includes the header line.

I was looking over this thread, always looking for tidbits of peoples ideas and thoughts. What I found interesting, the two k values that produced a low # of candidates after the sieve, but also only a few primes, if you factor the k-1, give a 2,3,7 and x. I tried this with k = 2940043 and found also a very low number of primes. What I did like, is that the bulk of the candidates get eliminated very quickly during the sieve phase.

That brings on the other major question, is it better to go for a k value that has a low number of primes but get sieved very quickly leaving a high possibility that the few remaining candidates yields a prime? or go along the route of the 15k where although there are a high number of primes, there are also a lot more candidates that don't get eliminated during the sieve phase?:question:

Regards
Patrick

That's the question alright! It probably depends on what you're
looking for. If you're looking for very large primes, you usually
use low-weight k's (the ones with fewer candidates). If it's too low,
you may not find any primes though. The 3rd drive here concentrated
on low-weight k's. Most of the other drives here have concentrated
on higher-weight k's, hoping for more primes.

I looked at some of the k values and characteristics of the exponents where their primes where found, in summary:

[CODE] 3^1 mix
3^2 odd
3^3 mix
3^4 odd
3^5 mix
3^6 odd
5^1 even
5^2 odd
5^3 even
5^4 odd
7^1 odd
7^2 odd
7^3 odd
3*5 mix
3*7 mix
5*7 even
3*5*7 mix[/CODE]

So it looks like a bit of homework could also eliminate a bunch of candidates beforehand.

A further bit of playing around, all the exponent values gave the following results:

k=7^1 mod 6 is either 3 or 5 apart from the first one.
k=7^1 mod 8 is either 1 or 5.
k=7^1 mod 12 is either 9 or 5 apart from the first one.
k=7^2 mod 6 is either 1 or 5.
k=7^2 mod 8 is either 3 or 7.
k=7^2 mod 12 is either 7 or 11.

I found similarities with the other k values as well.

Regards
Patrick