Very high weight k

[robert44444uk]

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

[kuratkull]

What program did you use to generate the Payam numbers? Sorry for being a newbie :)
Thanks

[kuratkull]

Bump! :)

[lsoule]

Payam numbers are described at
http://mathworld.wolfram.com/PayamNumber.html

[robert44444uk]

Originally Posted by kuratkull

What program did you use to generate the Payam numbers? Sorry for being a newbie :)
Thanks

I use a dos programme developed by Axn1, who has now left the mersenne group. I feel a little nervous about posting the software to this site without his specific permission, given the hard work he put into this. The programme is sensational in terms of speed and efficiency.

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.

[axn]

Originally Posted by robert44444uk

I use a dos programme developed by Axn1, who has now left the mersenne group. I feel a little nervous about posting the software to this site without his specific permission, given the hard work he put into this. The programme is sensational in terms of speed and efficiency.

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.

I'm still here (though at reduced activity levels).

Please go ahead and post the program here (warts and all ).

[lsoule]

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

[robert44444uk]

Originally Posted by axn1

I'm still here (though at reduced activity levels).

Please go ahead and post the program here (warts and all ).

Will post in a couple of days, past midnight here and I have a flight to catch early in the morning.

[robert44444uk]

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 [p|m|t] <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.

[kuratkull]

Thanks mate :)

[R. Gerbicz]

k=3902050318553160345 gives 87 primes of the form k*2^n-1 up to n=10000.