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2021-08-05, 17:50   #13
mart_r

Dec 2008
you know...around...

80810 Posts
That playground is gigantonormous!

Updated and polished, just a little. Added "higher-order descendants" (page 19).
This is a "MS print to PDF" version, while the one in the OP was a "FreePDF print" version; each version comes with its own minor glitches. If I'm ever able to convert it to proper LaTeX, maybe next year/decade, this issue will be taken care of.

After 1987#, the number of surviving primes n* in table 7 appears to be fixed for p<=601. The following level p=607 contains a split with a branch still present at level 1987, but dissipating at level 2089, so there are only 71 surviving primes at stage 111. You can see the "trajectories" for these weakest branches in the attachment. That split branch at level 607 started out like a flash in the pan! It's like a prime number stock market out there...

Quote:
 Originally Posted by mart_r I would expect one more prime to be cancelled out at level 797, leaving 225 "stable" primes at this point.
Down to 223 now. There's still a >1% chance it reaches 222, but <1% that the number drops even below that.

Supplemental tool for the backtracking process (only semi-general-purpose):
Code:
print("backtrack(b[,c]): track level b back to level c [or every level with < 512 surviving primes]");
backtrack(b,c)=
{
p=b;
t=p;
o=vector(primepi(p)-40);
n=#d;
o[1]=n;
k=1;
while(n>1&&p>c,
z=vector(512);
y=1;x=1;
for(i=2,#d,
if(d[i]<t,
if(d[i]>t/p,n--);y++;if(i==#d,z[x]=y;x++),
z[x]=y;y=1;if(x<512,x++)
)
);
k++;
o[k]=n;
p=precprime(p-2);
t*=p;
print("backtrack... p="p);
if(n<512&&(p==c||c==0),
write("backtrack_"b"-"p".txt",vecextract(z,Str("1.."x-1)))
)
);
o=vecextract(o,Str("1.."k));
forstep(i=#o,1,-1,
write("backtrack_"b".txt",o[i])
);
print("data stored in backtrack_{p-c}.txt");
}
Example: backtrack(1987,197) = "[96637, 113792, 156942]" - meaning: 3 surviving branches at level 197 with respectively 96637, 113792, 156942 primes at level 1987. (I'll upload the level 1987 data depending on the resonance, since it has 3.74 MB zipped.)

Additional Table 7 info not really fit for the paper:
The number of surviving primes n* appears to have settled for...
Code:
s = 30 to 36  after s = 48
s = 37 to 65  after s = 95
s = 66 to 70  after s = 162
s = 71 to 92  after s = 189
s = 93        after s = 191
s = 94 to 108 after s = 215
s =109 to 110 after s = 277
s =111 to ??? after s = 316
(s=139        after s>= 365)
So, when I give you these 65 primes of level 599 (s=109), you'll (hopefully) end up with all primes of the sequence after level 1787 (s=277):
Code:
a=39624069013936965087096009433866594830980870998888700289477858045283065746616452803046877051058323608232360514007056743887671907849763652319988387476334527612284794568776090626036384425467238802263863077912348447009215427300887220486657625693147;
d=[0, 466455760294422823953804039675640047201077790734154654780965233768051077060385506551407568084738361988997027851963959610818647415718916847152, 44895563831368704616159353930540997828272453622716098834658210530168205692523959883757599112304579276962544182213994859302153464025610, 26706100635873975723434507798935828965926546552594370, 5349424105738534, 901898922830014190586001959240919331648326160928257099022394892456073146632010, 8995009362764739858959068329200056018893114699905074080917303621788351313861729098338404578, 335544104419153579655358597659804721070678961457899223253722433765243412023981115104923784391687635420326417875238, 4523471714186028497063758655500264647233710442107467015046892971714017309602121083966315477490178979402201052, 13558128246316966557562438271899907310750726822, 4934323886885543883827676375700166702424555062100227255395720919371207047219908212443254804028675171530193258025833342400752532251499752430581830, 388743572956706723444269225383396603443044134640778117599062700920205080348, 7131662087452, 33194405695790798430238872743598395294, 269730, 135320865050066733911728570, 11221001162, 6090567717995320705159336935158027964523675153732520176913426535613320656808731739785868047152936659580764487743815834176652322675806820457799664407012544927967378642, 966178811730010342231796148, 950218907960867236355409812416967381787168349447008, 186, 120950, 73131703534393817971440867435305555941325599875579554494296006281538000190675946442663191144668123588, 9500734976302631650958, 239805326438620473888522605578, 121757244991625966205374175349841785534615242815048191144254, 66687483923550353133932113715210392764921421988484986752527792068, 55963473253722215455307053540553065129204623773008139755208417335670283622191194014140, 374220210121664223149161671575248005142427307251864790690209608349390524597485609165745496013651149975393673161307792844136375404294244730453984728505635072279055652, 1209221770576241494563186529310, 28686868, 513195473075718020212004909660243183839001052209643336479222370958923901013785264992668591798528873698138934, 28832971082846920, 48533460370249984502739595104518, 41532609059904971662727814316530245330921320936385322890063242134040854418808228566236415441515393074907863930376015390, 493668696637110236477409740705170, 1808806165895014602770, 2261804119760347690745179746, 9311195034, 99434466581329648062553784651274547675969070085859370489887908418, 11225742884389569244962733297207525945870303941562175609336782988922, 112530771834007266493561947078591371253972340193864418633152934638854286685371266094730252277266033265826678554646900216277609060540241061184441929856119915131915176, 103350, 44740191674963654791540507699261670144839100857418098222434989798279884602883087809574557987137117510760, 9254712663124863672244531853950613218514667190892855782913340344365740049591921814953513360801840244416196898101508293163612904791809371838669710474565685029165706842604, 10033871640846413419993982882804977302146190667092951612473356704835668, 773645088515448882592365176248227130967319190028631391234006175499055409359482, 6309191045489857456582839123180771159412906264449006658980416499498, 805748915562249520203749088487780366455469723998114574728718044699745578442712351844994056568611864541729304569354505304688218, 1871911391108669620366209771942252280517428081404, 281083455286058273331808398475392, 4187509955264496840236, 10812133799323198, 9428157516571371362264424890339613308558429717606124684073379249324365172732443776189794620110086458638386, 203548, 40180857345840, 2060098878522283566900456084770695449091282256414174589864543021569432, 670291401847140231046228944641480671586531001527350972352877129757263171304541394348129386594411256, 2452831165494447922493076252, 71305177701894194705286338516277103466839112851099161326083402585320861927804314198838482177449447105067911900878227962, 158419166957241279541816714465231159780430892544161477518866140, 1284912443358386721676929857796019465275702641192465355368497364626176221916, 17223504, 318, 154391504732234881359906941526553166298281939591864102142977844101440097198716489216556904069919213710];
I'm having way too much fun with this crazy sequence...
(Then again, some say I'm also having way too much fun watching cartoon shows...)
Attached Thumbnails

Attached Files
 Project p#Y.pdf (1.02 MB, 140 views)