20070331, 07:33  #1 
Jun 2003
Oxford, UK
3615_{8} Posts 
Very high weight k
I generated some E52 Payam Numbers and then ran the first 30,000 or so through the Nash program, and for k < 2*10^18 the best results were:
1540289647689526305 8321 8325 104638339734650865 8336 8323 553356619963699695 8282 8293 372349639969432125 8279 8282 303443427206348715 8296 8281 596126609590333965 8262 8269 211795368487356225 8261 8269 893386008747621675 8258 8255 541726696764825 8247 8245 319257076933318095 8221 8238 1207387684290870225 8237 8231 545866456597997535 8241 8231 1824593579944198635 8229 8228 1793052151130579625 8213 8221 1562451592176289815 8229 8221 264767258358587985 8226 8218 199103567327900235 8208 8210 1227114671166403305 8202 8207 810216647861986365 8196 8206 1743046040807216145 8191 8205 578295298798646715 8207 8203 These are some of the highest Nash weights I have come across, however this does not necessarily imply a lot of primes. I ran 3 of the above up to n=10000 and my results were: 104638339734650865 67 primes 553356619963699695 84 primes 541726696764825 66 primes A good Payam E(52) number has 100 primes in the first 4000 n. Regards Robert Smith 
20070401, 09:49  #2 
Mar 2007
Estonia
2·71 Posts 
What program did you use to generate the Payam numbers? Sorry for being a newbie :)
Thanks 
20070403, 15:23  #3 
Mar 2007
Estonia
2×71 Posts 
Bump! :)

20070403, 21:51  #4 
Nov 2004
California
2^{3}·3·71 Posts 
Payam numbers are described at
http://mathworld.wolfram.com/PayamNumber.html 
20070404, 05:23  #5  
Jun 2003
Oxford, UK
1,933 Posts 
Quote:
Perhaps if someone else knows how to contact him, and asks that specific question "can we post the software to the mersenneforum site?" I would be happy to oblige. 

20070404, 08:18  #6  
Jun 2003
4,969 Posts 
Quote:
Please go ahead and post the program here (warts and all ). 

20070404, 17:11  #7 
Nov 2004
California
2^{3}×3×71 Posts 
Just for fun, here are the other prime counts for the other k's up to n=10k
79 199103567327900235 73 1793052151130579625 72 1824593579944198635 72 1540289647689526305 69 596126609590333965 69 545866456597997535 67 1207387684290870225 67 104638339734650865 66 578295298798646715 66 211795368487356225 65 893386008747621675 65 1227114671166403305 64 303443427206348715 64 1562451592176289815 60 372349639969432125 60 264767258358587985 57 810216647861986365 54 1743046040807216145 53 319257076933318095 
20070404, 18:08  #8 
Jun 2003
Oxford, UK
1,933 Posts 

20070407, 08:49  #9 
Jun 2003
Oxford, UK
1933_{10} Posts 
Payam number generator
Back from my trip, here is Axn1's Payam number generator. This is a windows executable, and from the subdirectory you should use the instruction:
payamx [pmt] <e> <start> <end> {<cc> <ee>} p is a plus as in an E+<e> sieve of k2^n+1 m is for the minus one case t is both plus and minus, i.e. will find numbers which have no small factors in both the plus and minus case Start and end are to be written out in full i.e. 1000000000000000 "start" should be less than 10^19, and the end value should be less than ("start"+10^15) e is the payam level (max 256) cc, ee are optional secondary sieve. It sieves the survivors on 1<=n<=cc for all p with order <=ee This is useful for bitwins AS an example, the command payamx m 52 1 1000000000000 will produce the first few E(52) payams in the minus series. 
20070407, 14:41  #10 
Mar 2007
Estonia
2×71 Posts 
Thanks mate :)

20070407, 20:12  #11 
"Robert Gerbicz"
Oct 2005
Hungary
1468_{10} Posts 
k=3902050318553160345 gives 87 primes of the form k*2^n1 up to n=10000.

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