[QUOTE=Random Poster;223943]if p and q1, q2,... are primes (qi not necessarily distinct) and p divides qi for all i,[/QUOTE]

I think something got left out. p divides qi, qi a prime?

[quote=wblipp;224086]I think something got left out. p divides qi, qi a prime?[/quote]
Oops. I meant "p divides qi+1".

[QUOTE=Random Poster(fixed);223943]There certainly are n-level factorizations for any n; if p and q[sub]1[/sub], q[sub]2[/sub],... are primes (q[sub]i[/sub] not necessarily distinct) and p divides q[sub]i[/sub]+1 for all i, then Φ[sub]p[/sub](Φ[sub]q1[/sub](Φ[sub]q2[/sub](x))) has three factors, Φ[sub]p[/sub](Φ[sub]q1[/sub](Φ[sub]q2[/sub](Φ[sub]q3[/sub](x)))) has four factors and so on.[/QUOTE]

I was a little slow, but I see this now. The new factor at each level comes because Φ[sub]q[/sub](x) evaluated at plus or minus a p[sup]th[/sup] primitive root of unity is minus or plus another p[sup]th[/sup] primitive root of unity. Hence the roots of Φ[sub]p[/sub](x) or Φ[sub]p[/sub](-x) are also roots of entire chain, depending on whether the nesting level is even or odd. The old factors at each level come from evaluating the innermost level y=Φ[sub]q[/sub](x), creating a shorter nested chain in y.

Looking at WraithX's tests for odd primes p and q (the ones OddPerfect is most often interested in), we see all of these factors and no other factors.

William

From Pascal's [URL="http://www.lri.fr/~ochem/opn/t600.txt"]t600[/URL] and the [URL="http://oddperfect.org/composites.html"]OddPerfect Composites Page[/URL], factored by Anand Nair

[code]
C139 from sigma(33579221106357041^10) = P60 * P79
P60: 947684437767587328462115861183376363741875478272499305419101
P79: 6052327488685215057461061918517178500155472853109995698461576756862437371530709[/code]

From Pascal's [URL="http://www.lri.fr/~ochem/opn/t600.txt"]t600[/URL]

ECM to t50 by yoyo@home
SNFS sieving by RSALS
Post Processing by Carlos Pinho

[code]
sigma(331^82)

P77: 12253931044501974347843705546022379251172212072383838609763792567641647710229
P78: 959481379440808778268344387753224454423782269330139108178791067284341494690161
[/code]

From Pascal's t1200 page:

[code]sigma(819361018009248711269490108424927578504639579897569773^2)

P64 = 8895901114861096166655445036853111562014397837878284573466205877
P33 = 162852231451550720161747459113637[/code]

Factored using YAFU, data:
[code]factoring 1448717347327467320748607108446865593378017750403357902354464511952856354567613687050595780244649

div: primes less than 10000
rho: x^2 + 1, doing 190000 iterations on C97
rho: x^2 + 3, doing 190000 iterations on C97
rho: x^2 + 2, doing 190000 iterations on C97
pp1: doing B1 = 20K, B2 = 1M on C97
pp1: doing B1 = 20K, B2 = 1M on C97
pp1: doing B1 = 20K, B2 = 1M on C97
pm1: doing B1 = 100K, B2 = 5M on C97
ecm: 25 curves on C97 input, at B1 = 2K, B2 = 200K
ecm: 90 curves on C97 input, at B1 = 11K, B2 = 1100K
ecm: 200 curves on C97 input, at B1 = 50K, B2 = 5M
ecm: 291 curves on C97 input, at B1 = 250K, B2 = 25M
ecm: 9 curves on C97 input, at B1 = 1M, B2 = 100M

starting SIQS on c97: 1448717347327467320748607108446865593378017750403357902354464511952856354567613687050595780244649

==== sieving in progress ( 4 threads): 90742 relations needed ====
==== Press ctrl-c to abort and save state ====
91702 rels found: 24421 full + 67281 from 1226556 partial, (478.12 rels/sec)

SIQS elapsed time = 2686.6220 seconds.
Total factoring time = 3721.7070 seconds


***factors found***

PRP64 = 8895901114861096166655445036853111562014397837878284573466205877
PRP33 = 162852231451550720161747459113637

ans = 1[/code]

So this thread concerns factoring sigma(p^n), where p is prime?

Ex: sigma(503^76) = [code]20894459062678117658306604117618920160599312073908602533189948593605211532436014706409146910962835558801054213210058283311311962330790240068832980312102877669963441159845489564144460574295153629947339384881[/code]

Well, that left me with a c183 cofactor.

No, wait, I found a p20 factor. It's a c163 now. I'll treat it to some more ECM, to see if I can find any more small factors. If I'm lucky, the remaining cofactor should be easily factorable or, even more luckily, a PRP.

Or has this number already been factored previously?

Found a p30 factor via GMP-ECM, with B1 = 50k.

[code]********** Factor found in step 2: 178745360515311929424871408547
Found probable prime factor of 30 digits: 178745360515311929424871408547
Composite cofactor 6933939647989085278344851249440665011703817855287480955478872511024356491515718634487262962102752919558837937500813142743695583166107 has 133 digits[/code]

There's not much I can do about the c133 cofactor.

[quote=3.14159;224395] So this thread concerns factoring sigma(p^n), where p is prime?
[/quote]

Only those in Pascal's t1200.txt. And n will always be 1 less than a prime. So sigma(503^76) isn't interesting since 77 is composite.

To see if a number's already been factored look in checkfacts.txt.

Chris K

Also: How does this assist in the search for an odd perfect number?

Okay, disregard the above.

Reading the papers on Pascal's web site should tell you how factoring t1200.txt helps.

To contribute run the following:
awk < t1200.txt '{print $NF}' | ecm -nn 3e6 | tee -a t1200.ecm | grep '[Ff]actor'

It will take a few weeks though. But you can post results when they come out.

Chris K

Carlos Pinho has attacked the smallest composites of t1200 with YAFU. I have not determined in which t file each composite first appears.

[code]
sigma(142284955959247363491774930128758921742556021534481^2)
P43: 2238591570751691275025203984739064590104171
P51: 358263586731013950068031708899681908858371624877057

sigma(5405967912750465340092257739308075946643484287840514164962968461618901^2)
P49: 4581143948207056755757357143366580942127861508157
P45: 302952143649098442261312724375630723647869797

sigma(823272754340808326071857084594505120710944210886853003592361373428960205057919^2)
P44: 35193307933628882071541389173031929116191771
P52: 1417962540975410444369220209761826181413254300565971

sigma(31760490959072695223529283^10)
P43: 2492627611840943383968864765610539376675291
P54: 300350691953269947918894929057862181065708207619522979

sigma(269936558197762891^10)
P45: 908603472850015693984764072213114333991284361
P53: 28801923821297214583405169855730516656550971797022411

sigma(3141295094798804242081589115747839899155140002935202163479^2)
P26: 11318127709881100296762883
P28: 8709472493279809677543843709
P45: 646555076222326204672887005617748263663956703

sigma(8153965723441849497333245425664590336310402817099782160707512036971303217832758231^2)
P38: 48879638228126430861096072072409741879
P61: 1949113651453840360294368987837410363945120729248662107761327

sigma(9892750003972963203348605103578960684769474622272592786703078222796786625340526101^2)
P30: 921474140411983643888916057073
P69: 217158109183717276336393167849260577432700103824794467676798922133623

sigma(26003471145623291369238400209138039373^4)
P41: 54795424046653166474311338921067237111831
P58: 5111606179422695424324475676443070008706462534402777885161

sigma(631172722633094599282215430260850529796246854551094582836527975364552271665451^2)
P41: 29289589722341952820875974079864400161667
P59: 35613271398054634765430994608208717929829595048790195209363

sigma(10804085173076921971631847502541205383694345195411388477743561650435473^2)
P46: 1590620939691289032811391988951155376074673037
P54: 871087318367889270584056815223555941258522686821185877
[/code]