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1. Purpose of numbpart@trinitas.mju.ac.kr
[ ÇÑ±Û ¼³¸í
]
Automatic computation of the value of the partition function p(n) for a given number n,
using Rademacher's formula for p(n) and immediate return to the e-mail sender
of the computed result.
2. Introduction
We first recall that p(n) is, by definition, the number of writing a given
number n as a sum of others in a non-increasing way. For example, p(5) = 7,
since 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. At around 1937, Rademacher
discovered a very complicated but theoretically exact formula for p(n) in the
form of a convergent infinite series.
Rademacher's
Theorem for p(n)
We use his formula in this Service of our InetCompu. His formula is not
recursive and is, up to now, practically the only efficient formula allowing us to compute
the value of p(n) particularly when n is bigger than, say, 510,000. For
instance, if one tries the command, numbpart, within Maple, it will be impossible
for him to compute p(510,000) even when the size of available main memory is 128 Mbytes, because Maple's command, numbpart, is recursive and
hence it requires too much memory. Try to compute the value of p(n) when
n=100,000 with Maple that is installed in your machine or Server, in order to
see if it is even possible!
But, there are several problems in using Rademacher's formula for p(n) in actual computation, and
one of
serious problems of Rademacher's formula is - one has to rely on approximated computations of fractional real numbers of very much different sizes of which partial sum converges to not-yet-known true integer value of
p(n) for a given number n.
3. Contributor: The program we are using
for Rademacher's formula for p(n) was developed in Reduce
commands by Prof. Sun T. Soh, Dept of Math, Myong Ji Univ., Rep. of Korea.
He developed this program to celebrate lifetime
works of Prof. George Andrews at Penn. St Univ. on
Addtive Number Theory.
4. Notes:
4-1. His program is
very reliable and efficient. In fact, it can
compute the values of p(n) for all n such that n
<= 22,000,000 ~. Such an upper bound is
only limited by inability of the current version
of reduce in properly handling input/output
buffers, especially when the number of decimal
digits of data approaches very near to
10^5000.
4-2. For a trouble-free handling of your e-mail, we recommend you to use MicroSoft Outlook Express, New mail > Alt+O >
Alt+X (with No Encryption), to send out an e-mail to
us.
5. How to do: Send an e-mail with plain text style
(for instance, in the case of MicroSoft Outlook Express, New mail > Alt+o
> Alt+x (with No Encryption)) to numbpart@trinitas.mju.ac.kr
whose main body should consist of, for example,
input: n=1234567
Then upon the arrival of the e-mail, the value of p(1234567) is
automatically computed and sent back to the e-mail sender immediately.
[Reminder] When the requested job is computationally
not very complicated, it should be quite the case that you will receive the
result within a few Minutes. But, our response consisting of the computed
results can not be delivered to the sender properly, if either
there is a spelling mistake in sender's e-mail address or sender's mail box is
already filled with too many of other e-mails. Thus, if you do not receive the results although you
have waited for some time, then please check your mail account to correct the
above trouble-causing problems. After that, try again according to the
procedure described in Section 5 above.
6. An Example
If you send an e-mail to numbpart@trinitas.mju.ac.kr
whose main body consists of
input: n=1234567
then you should receive the following lines as output:
Comment The following is the requested input:
n:=1234567;
Comment Here is the result of computation:$
comment t(k) = the k-th term in Rademacher's formula for p(n) when n := 1234567$
t(1) :=
715490880908500090668830204971571483282076915621981638292949344880079431938962\
576616856283139608159603320390006454590993465575888598311877101185251260952638\
104333567814638303161155133579697248237093428793577819038063905854717015597399\
585216283540966035978582677467955302786975214400242521093365769853143215687615\
543827820468133192277814647650183125609480408170481114482606332411772207385499\
476998482655218827261186031705023313307874014273301300297053610135197709439985\
904984032185374946844419382113860443967309684571387268375077055167683611188310\
415800797515219481885264654498863641033307739629340318142909047961816717574545\
038063302180706901599594598480550744900310304080125051702741506545540809585926\
876061043777766937526852464833287846420247276056627723451320743239594568836087\
432035753579659123314630190205931905650212235380650496590914435629227555569797\
394845013526086367203950617996367530901222036993907458991864726450938183881600\
915881548287019564815637120286792310876464204116153233297878185629018841656811\
492995304925738482756784778728332435010508826332097457790418230853782763077548\
360783957283548013266304181056802001515254028085673569648455221323427253150692\
9143817463763141204784660574943985675960042312092398756262386.9437436834509$
t(2) := -
646384239560223084468700078738095411758409495443243448093736135967739258882037\
774527614475653437409318783504740805421610292215347524845105940560984611714986\
786623749874235920746618953914502376843040089031989394265073634257807077244404\
062352503105930587904264977702974512649074341050229731966175604233066030330234\
307406911681681257612172291356124526629568956906587894503947646784103577388980\
679126431730521704470046148001122592696950681684430913210551779367779461340209\
322424247155373016995996438462775669191018313758462216767191962451490325866372\
890617490148069677199837667644858301897020579862678726227528312154.43224937895\
289626087241816615742503234155822879046613613871682343315066147152909044535916\
906024468807661544821208988802453277436215206927921023661052398425720471980381\
186656214712329313724912565360091942684759308908398989058116187663004117493434\
550406247118803447618217561515426098117407396824086017201837399416605815410800\
671611760364900601446083131889788446904043469498265553408510457153360214808927\
791050228677220975390691458630333307662787529491078400913901859210397263395358\
130382346431862027367204140792889522864494160543847958997683652130796271083358\
94726808151860595791817573865577852076880266129281583234465969336811167393$
t(3) := -
181785400593037617890851877545890519094020758838796249145498356571083421061165\
098430182568600222590696691333940618327041936790907944356232309774007217402686\
072092782918146719250091774850108250597131815736721173938655810550922026404583\
130103620349489712271261765263412393535148217298250510846721377978369195408381\
123994418705839230256228597215235786660663726651069494084793307496328908494441\
9496044749300817.8922369923143297596496160441656509093800091717980493332763507\
066148332906189091605658933644323852267288380612012562387202450215380015555977\
422834132905177917299942642526842965627249334825102755343353512905921463457720\
6$
t(4) := -
124973345432793477526913218024113417024916155596319716123900959167935831005396\
172378408257248628952119954120827771902926635195618013278515505329438529964191\
820497465818303503321246571393565243703401860700054404240912760797087177747848\
808228621184547854673600854457728334300730549220657162677022911202661.47052287\
194938589447453633846882435831094510846161917003127791070676100195557686711602\
77671682001505642728029993835$
t(5) := 0$
t(6) :=
186238997239247843684104917101252915830122485133578995034766181136075336596004\
553993764863958521236375120902602085971551899688731898216825401411241711584224\
19541853496183638018799857080744964643891233.553683488467194339233853874863015\
683384188799771782342369985551455363359616337979504556822421726439688442342012\
7818$
t(7) := -
782520273130950940627774875109867272632972467666186471049681298447822193631719\
962723187428236919898967706007960079055845751830930524810709495156506495451752\
34028819784065.000861224202169752573014341824510568499276$
t(8) := -
604754911052939226456034391872992447704633570202965492427589981678301017516836\
2266770223794822400334715770801668512798250957201315293313641045366287.2924012\
566952941688303244127821482$
t(9) :=
531671160751248513632888911446446531950380552206448678891567846550022017258961\
9722996751778766707254670302614416212425287699988557.7585664016381091472487769\
35273$
t(10) := 0$
t(11) :=
324921599983802460692676060173448137333975531659562884202821463306518012302065\
7792394925249969620405958323.121107775976570607721068249856007803$
t(12) :=
112484373196395077295180066358650407543584937526839868782854081117429085958646\
5750905992364267318.561725300705727046801$
t(13) := 0$
t(14) :=
213088786218440868550405688355782398118118974493463188161881742914479812575674\
7331.586376957247901466071147513$
t(15) := 0$
t(16) :=
20667403333394079253304664685077788651865766621878024093070894329661205.506789\
38971150886421281$
t(17) :=
906450859895924560495888498222650293648359950333177148016386935844.21941872216\
689439$
t(18) := -
55242159793777147410364343715233148256990306709665248559781550.9058920657701547$
t(19) := -
16279399009539245745659540870001333108615235753769869094725.055062922133127$
t(20) := 0$
t(21) := -
11553317602807885324795961579138538164936815762289976.814389083214838584$
t(22) := 25291922427566184313788067346969907912398651177922.81162436303916$
t(23) := 0$
t(24) := - 95478594383391793803180919232259944913149870.350114670789738753$
t(25) := 0$
t(26) := 0$
t(27) := - 789809998825346369987261104083803489684.5516839650229179$
t(28) := - 7130030931856561784529910339177479075.47136216872206$
t(29) := 0$
t(30) := 0$
t(31) := - 681500048611159977143858036826989.1192143954378274$
t(32) := 23702464396157510272556638243492.5767429429156$
t(33) := - 1206706386657301981151550891981.4649828812127$
t(34) := 185244168637809133888016876576.3977649815244$
t(35) := 0$
t(36) := 896148501102143827780865649.056507576163826$
t(37) := - 473620382027359942712402247.869064826654$
t(38) := - 30539341852964064313180623.7654107381355$
t(39) := 0$
t(40) := 0$
t(41) := 0$
t(42) := 17887743689306242195180.505846212286179$
t(43) := 6429550741850582916459.4000079441379$
t(44) := - 2762582195494791384703.150433521969$
t(45) := 0$
t(46) := 0$
t(47) := 0$
t(48) := - 2324663733836091895.423179070150919$
t(49) := 0$
t(50) := 0$
t(51) := 124897864676152106.9695564614487$
t(52) := 0$
t(53) := - 39318902702708395.6621000656839$
t(54) := - 3780415831433575.6084029456651$
t(55) := 0$
t(56) := - 283674101024237.5063807196981$
t(57) := 235056197520887.480187107817$
t(58) := 0$
t(59) := - 115842930553315.016749655118$
t(60) := 0$
t(61) := - 35113131351459.1468425893897$
t(62) := 12227867395983.685302776678$
t(63) := 2630810836497.4682597355395$
t(64) := 2441961456648.624086192557$
t(65) := 0$
t(66) := - 445899676621.3883196494486$
t(67) := - 349553228850.989603702593$
t(68) := - 118447343337.147079034207$
t(69) := 0$
t(70) := 0$
t(71) := 0$
t(72) := 3663423735.03960176611819$
t(73) := 20511637850.77847211184$
t(74) := 3201447309.0065847618363$
t(75) := 0$
t(76) := 2034284831.885585389844$
t(77) := 2595353016.901182996317$
t(78) := 0$
t(79) := 0$
t(80) := 0$
t(81) := - 246295392.7878120033387$
t(82) := 0$
t(83) := - 31501914.9348508063765$
t(84) := 22519937.938630167419$
t(85) := 0$
t(86) := - 20641072.174122690619$
t(87) := 0$
t(88) := 3119828.8744769323185$
t(89) := 0$
t(90) := 0$
t(91) := 0$
t(92) := 0$
t(93) := 3146705.55521874493$
t(94) := 0$
t(95) := 0$
t(96) := 96306.37431131007617$
t(97) := 0$
t(98) := 0$
t(99) := - 383539.42933857271$
t(100) := 0$
t(101) := 306468.661337514042$
t(102) := - 230767.453571350339$
t(103) := 0$
t(104) := 0$
t(105) := 0$
t(106) := - 70972.2393080120499$
t(107) := 0$
t(108) := 33590.209434412711$
t(109) := - 2205.3724639855506$
t(110) := 0$
t(111) := 12166.12897397441$
t(112) := - 12590.018778048684$
t(113) := 0$
t(114) := 6504.6246847185273$
t(115) := 0$
t(116) := 0$
t(117) := 0$
t(118) := 2371.783733219912$
t(119) := 521.1894471713266$
t(120) := 0$
t(121) := - 2772.23360602359$
t(122) := 727.387074013241$
t(123) := 0$
t(124) := - 783.754878550522$
t(125) := 0$
t(126) := 140.209293312099$
t(127) := 1081.46194554012$
t(128) := - 407.9658015343132$
t(129) := 933.687658234734$
t(130) := 0$
t(131) := 0$
t(132) := 33.3669457466735$
t(133) := 225.82846750291$
t(134) := 236.675651959201$
t(135) := 0$
t(136) := 75.4787604700645$
t(137) := - 186.36789146153$
t(138) := 0$
t(139) := 0$
t(140) := 0$
t(141) := 0$
t(142) := 0$
t(143) := 0$
t(144) := - 2.72167989843604$
t(145) := 0$
t(146) := 47.1121121926293$
t(147) := 0$
t(148) := - 39.180862184586$
t(149) := - 29.898783160802$
t(150) := 0$
t(151) := 0$
t(152) := 8.553355865012$
t(153) := 6.90190020446$
t(154) := 2.4270745418037$
t(155) := 0$
t(156) := 0$
t(157) := 0$
t(158) := 0$
t(159) := 5.1639892210306$
t(160) := 0$
t(161) := 0$
t(162) := - 3.0315068434988$
t(163) := 0$
t(164) := 0$
t(165) := 0$
t(166) := - 2.8709093900987$
t(167) := 0$
t(168) := 2.4261967611585$
t(169) := 0$
t(170) := 0$
t(171) := 2.0180306676496$
t(172) := 2.7826165778986$
t(173) := 0$
t(174) := 0$
t(175) := 0$
t(176) := - 0.73051301766582$
t(177) := - 0.29688077130683$
t(178) := 0$
t(179) := 0$
t(180) := 0$
t(181) := 0$
t(182) := 0$
t(183) := 0.98130095750553$
t(184) := 0$
t(185) := 0$
t(186) := 0.27000839085032$
t(187) := 1.1102436539792$
t(188) := 0$
t(189) := 0.41638953115812$
t(190) := 0$
t(191) := 0.53495973448487$
t(192) := - 0.12015047704749$
t(193) := - 0.41988633092102$
t(194) := 0$
t(195) := 0$
t(196) := 0$
t(197) := 0.10222435860785$
t(198) := 0.29136038804454$
t(199) := - 0.17220792647499$
t(200) := 0$
t(201) := - 0.17707571548646$
t(202) := 0.072166712944328$
t(203) := 0$
t(204) := 0.17096910659204$
t(205) := 0$
t(206) := 0$
t(207) := 0$
t(208) := 0$
t(209) := 0.025792598279001$
t(210) := 0$
t(211) := 0$
t(212) := 0.13572367635812$
t(213) := 0$
t(214) := 0$
t(215) := 0$
t(216) := 0.034168428975676$
t(217) := - 0.0055355590174119$
t(218) := 0.050307742504395$
t(219) := 0.018169248768499$
comment The partial sum up to the 218-th term is: $
Rademacher(1234567) :=
715490880908500090668830204971571483282076915621981638292949344880079431938962\
576616856283139608159603320390006454590993465575888598311877101185251260952638\
104333567814638303161155133579697248237093428793577819038063905854717015597399\
585216283540966035978582677467955302786975214400242521093365769853143215687615\
543827820468133192277814647650183125609480408170481114482606332411772207385499\
476998482655218827261186031705023313307874014273301300297053610135197709439985\
904984032185374946844419382113860443967309684571387268375077055167683611188310\
415800797515219481885264654498863641033307739629340318142909047961816717509906\
614107279872260031591720788939374903950765979735315678089144732619652605808474\
114613478434026005648501990752745685391025741304143212857264644778423070157425\
057048329987584461419238739968247601641309036259438588634451037030651237617671\
625440399708457144457034822188531540046055848922280658862841197257208456619585\
063106990471776117839733933821863731203738887642219959698259472920648551500949\
773500258044342763418353205860697591237423169787525187299963229569479391673799\
046398119149345658525737875496420744725360387951929243107396970737312263130169\
8197398456788392768874673519277489020839294797279106528252500.0027065859747$
comment The partial sum up to the 217-th term is: $
Rademacher(1234567) :=
715490880908500090668830204971571483282076915621981638292949344880079431938962\
576616856283139608159603320390006454590993465575888598311877101185251260952638\
104333567814638303161155133579697248237093428793577819038063905854717015597399\
585216283540966035978582677467955302786975214400242521093365769853143215687615\
543827820468133192277814647650183125609480408170481114482606332411772207385499\
476998482655218827261186031705023313307874014273301300297053610135197709439985\
904984032185374946844419382113860443967309684571387268375077055167683611188310\
415800797515219481885264654498863641033307739629340318142909047961816717509906\
614107279872260031591720788939374903950765979735315678089144732619652605808474\
114613478434026005648501990752745685391025741304143212857264644778423070157425\
057048329987584461419238739968247601641309036259438588634451037030651237617671\
625440399708457144457034822188531540046055848922280658862841197257208456619585\
063106990471776117839733933821863731203738887642219959698259472920648551500949\
773500258044342763418353205860697591237423169787525187299963229569479391673799\
046398119149345658525737875496420744725360387951929243107396970737312263130169\
8197398456788392768874673519277489020839294797279106528252499.9523988434703$
comment The partial sum up to the 216-th term is: $
Rademacher(1234567) :=
715490880908500090668830204971571483282076915621981638292949344880079431938962\
576616856283139608159603320390006454590993465575888598311877101185251260952638\
104333567814638303161155133579697248237093428793577819038063905854717015597399\
585216283540966035978582677467955302786975214400242521093365769853143215687615\
543827820468133192277814647650183125609480408170481114482606332411772207385499\
476998482655218827261186031705023313307874014273301300297053610135197709439985\
904984032185374946844419382113860443967309684571387268375077055167683611188310\
415800797515219481885264654498863641033307739629340318142909047961816717509906\
614107279872260031591720788939374903950765979735315678089144732619652605808474\
114613478434026005648501990752745685391025741304143212857264644778423070157425\
057048329987584461419238739968247601641309036259438588634451037030651237617671\
625440399708457144457034822188531540046055848922280658862841197257208456619585\
063106990471776117839733933821863731203738887642219959698259472920648551500949\
773500258044342763418353205860697591237423169787525187299963229569479391673799\
046398119149345658525737875496420744725360387951929243107396970737312263130169\
8197398456788392768874673519277489020839294797279106528252499.9579344024877$
comment The partial sum up to the 219-th term is: $
Rademacher(1234567) :=
715490880908500090668830204971571483282076915621981638292949344880079431938962\
576616856283139608159603320390006454590993465575888598311877101185251260952638\
104333567814638303161155133579697248237093428793577819038063905854717015597399\
585216283540966035978582677467955302786975214400242521093365769853143215687615\
543827820468133192277814647650183125609480408170481114482606332411772207385499\
476998482655218827261186031705023313307874014273301300297053610135197709439985\
904984032185374946844419382113860443967309684571387268375077055167683611188310\
415800797515219481885264654498863641033307739629340318142909047961816717509906\
614107279872260031591720788939374903950765979735315678089144732619652605808474\
114613478434026005648501990752745685391025741304143212857264644778423070157425\
057048329987584461419238739968247601641309036259438588634451037030651237617671\
625440399708457144457034822188531540046055848922280658862841197257208456619585\
063106990471776117839733933821863731203738887642219959698259472920648551500949\
773500258044342763418353205860697591237423169787525187299963229569479391673799\
046398119149345658525737875496420744725360387951929243107396970737312263130169\
8197398456788392768874673519277489020839294797279106528252500.0208758347418$
comment with the first 1-th fractional digits := 0$
comment Thus, one concludes that the exact value of p(1234567) is: $
p(1234567) :=
715490880908500090668830204971571483282076915621981638292949344880079431938962\
576616856283139608159603320390006454590993465575888598311877101185251260952638\
104333567814638303161155133579697248237093428793577819038063905854717015597399\
585216283540966035978582677467955302786975214400242521093365769853143215687615\
543827820468133192277814647650183125609480408170481114482606332411772207385499\
476998482655218827261186031705023313307874014273301300297053610135197709439985\
904984032185374946844419382113860443967309684571387268375077055167683611188310\
415800797515219481885264654498863641033307739629340318142909047961816717509906\
614107279872260031591720788939374903950765979735315678089144732619652605808474\
114613478434026005648501990752745685391025741304143212857264644778423070157425\
057048329987584461419238739968247601641309036259438588634451037030651237617671\
625440399708457144457034822188531540046055848922280658862841197257208456619585\
063106990471776117839733933821863731203738887642219959698259472920648551500949\
773500258044342763418353205860697591237423169787525187299963229569479391673799\
046398119149345658525737875496420744725360387951929243107396970737312263130169\
8197398456788392768874673519277489020839294797279106528252500$
comment The following is the timex report of the system in seconds:
real 36.34
user 35.74
sys 0.26
$
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