Ramanujan Type Congruences for the Number of Colored Integer Partitions

         with an Odd Number of Colors Up To, 47, and Moduli up to, 200



                              By Shalosh B. Ekhad



              Let , p_[-a](n), defined by the generating function

                  infinity                 infinity
                   -----                   --------'
                    \                 n   '  |  |        1
                     )     p_[-a](n) q  =    |  |    ---------
                    /                        |  |          i a
                   -----                     |  |    (1 - q )
                   n = 0                    i = 1

                     --------------------------------------

                         Fact Number, 1, : There are, 3

               Congruences for, 1, colors modulo primes up to, 200



         1/5 p[-1](5 n + 4), are all integers, the first, 5, terms are:

                              [1, 6, 27, 98, 315]



         1/7 p[-1](7 n + 5), are all integers, the first, 5, terms are:

                             [1, 11, 70, 348, 1449]



        1/11 p[-1](11 n + 6), are all integers, the first, 5, terms are:

                           [1, 27, 338, 2835, 18566]

                     --------------------------------------

                         Fact Number, 2, : There are, 2

               Congruences for, 3, colors modulo primes up to, 200



        1/11 p[-3](11 n + 7), are all integers, the first, 5, terms are:

                   [39, 12979, 1063818, 43747800, 1164428478]



       1/17 p[-3](17 n + 15), are all integers, the first, 5, terms are:

             [2093, 2002023, 425689641, 40914135699, 2350811217042]

                     --------------------------------------

                         Fact Number, 3, : There are, 2

               Congruences for, 5, colors modulo primes up to, 200



        1/11 p[-5](11 n + 8), are all integers, the first, 5, terms are:

              [615, 850770, 223786351, 25270630495, 1662147139125]



        1/23 p[-5](23 n + 5), are all integers, the first, 5, terms are:

       [22, 41929385, 553001147255, 1187595068587015, 901105728446782065]

                     --------------------------------------

                         Fact Number, 4, : There are, 2

               Congruences for, 7, colors modulo primes up to, 200



        1/11 p[-7](11 n + 9), are all integers, the first, 5, terms are:

         [7315, 27515173, 17210982243, 4187796348205, 552552745379845]



        1/19 p[-7](19 n + 9), are all integers, the first, 5, terms are:

  [4235, 1929783430, 23764157017930, 58690206350940353, 54645643050406314151]

                     --------------------------------------

                         Fact Number, 5, : There are, 3

               Congruences for, 9, colors modulo primes up to, 200



       1/17 p[-9](17 n + 11), are all integers, the first, 5, terms are:

[127305, 42942051360, 635196036306389, 2049324191739313434,

    2542703224855425352335]



       1/19 p[-9](19 n + 17), are all integers, the first, 5, terms are:

[18186957, 4516034722715, 75622674383356572, 303132076373508180615,

    480165308667925390850730]



        1/23 p[-9](23 n + 9), are all integers, the first, 5, terms are:

[13715, 367195036617, 62470904925381516, 1261880556758346655890,

    7150622715567528809798070]

Fact Number, 6, : There are no Congruences for, 11, colors with moduli up to ,

    200

                     --------------------------------------

                         Fact Number, 7, : There are, 3

              Congruences for, 13, colors modulo primes up to, 200



       1/17 p[-13](17 n + 14), are all integers, the first, 5, terms are:

[34998418, 47800169858077, 2885120019631813873, 34120619571733966682273,

    140786889041449714454220713]



       1/19 p[-13](19 n + 14), are all integers, the first, 5, terms are:

[31314374, 175561330794564, 26390955885841707293, 637941932763473776412897,

    4816130196905272955386991769]



       1/23 p[-13](23 n + 13), are all integers, the first, 5, terms are:

[9430568, 1127601900356061, 1060859012579403383346, 104059874508897615031380527,

    2526635717904843708246386826653]

                     --------------------------------------

                         Fact Number, 8, : There are, 2

              Congruences for, 15, colors modulo primes up to, 200



       1/23 p[-15](23 n + 15), are all integers, the first, 5, terms are:

[252041059, 51580407738099555, 95125691893239606701970,

    17759465244963941816551686500, 788190201810809144384547739194270]



       1/29 p[-15](29 n + 26), are all integers, the first, 5, terms are:

[4675667147235, 2300374814623794121941, 14085093125316229716575475500,

    8524132392672537505421873944817925,

    1151953310250788341129431072611025410415]

                     --------------------------------------

                         Fact Number, 9, : There are, 1

              Congruences for, 17, colors modulo primes up to, 200



       1/23 p[-17](23 n + 17), are all integers, the first, 5, terms are:

[6793517070, 2173837443110774663, 7212031302876156959478775,

    2387320575685516961457050470466, 182250530957089872545693999478000020]

                     --------------------------------------

                        Fact Number, 10, : There are, 1

              Congruences for, 19, colors modulo primes up to, 200



       1/23 p[-19](23 n + 19), are all integers, the first, 5, terms are:

[184333754550, 86106128826711882427, 479626383951997042457280783,

    265636559478597945089962381868210, 33178503911587201146061067742270048428]

                     --------------------------------------

                        Fact Number, 11, : There are, 3

              Congruences for, 21, colors modulo primes up to, 200



       1/29 p[-21](29 n + 19), are all integers, the first, 5, terms are:

[474643340325, 52074455451408729335143, 10954822710694259394919156433940,

    109992990517625172712516922279453354076,

    163835231234303492540632079554579766282465723]



       1/31 p[-21](31 n + 28), are all integers, the first, 5, terms are:

[3598596647882379, 112826228084213425628195475,

    18814511845764908569864434764698318,

    195016921190419264031205207418687263987684,

    332516585050492869398687032828848143890385851950]



       1/47 p[-21](47 n + 42), are all integers, the first, 5, terms are:

[338596997583178656355, 7088424037453018097218450688838975,

    160741537936542193217127783256759319663092830,

    102160821364197757224314997500444854045007676609122882,

    6466822954847092632647967799347122469195113491307088188419884]

Fact Number, 12, : There are no Congruences for, 23, colors with moduli up to ,

    200

                     --------------------------------------

                        Fact Number, 13, : There are, 2

              Congruences for, 25, colors modulo primes up to, 200



       1/29 p[-25](29 n + 24), are all integers, the first, 5, terms are:

[1044173468026000, 163586635252961971970481975,

    82874209979961437061257067932613425,

    2123535869640739363473014183421375560632600,

    7943880271506908216801979108278138348995657728345]



       1/31 p[-25](31 n + 23), are all integers, the first, 5, terms are:

[333381777587950, 327668365705225364064012775,

    507200134546077589593145333226124465,

    30380795522374983596673159437838117332850775,

    229889735910202411807003909081842031959301967702525]

                     --------------------------------------

                        Fact Number, 14, : There are, 1

              Congruences for, 27, colors modulo primes up to, 200



       1/41 p[-27](41 n + 37), are all integers, the first, 5, terms are:

[1075515526898331981210, 65035442331705077255565765231770970,

    3887632022555600351237636033836883253630795450,

    5984518541571582781521145502888791517097930193519811904,

    858133376064628881752091865606958393756743168542721305529418841]

Fact Number, 15, : There are no Congruences for, 29, colors with moduli up to ,

    200

Fact Number, 16, : There are no Congruences for, 31, colors with moduli up to ,

    200

                     --------------------------------------

                        Fact Number, 17, : There are, 4

              Congruences for, 33, colors modulo primes up to, 200



       1/41 p[-33](41 n + 27), are all integers, the first, 5, terms are:

[1724184437187745610, 47980710790857123081236259384348504,

    120047235893679494383254701292123683128146588720,

    3237569571657625125383603283877099520359128249530568840323,

    5092876482851207768845291788064086353661873140663913688104778236225]



       1/43 p[-33](43 n + 39), are all integers, the first, 5, terms are:

[536254820831772058683326, 1555766448504867219696225584190731277435,

    2042242670437295890578859672472217918867454589638014,

    44944931006181833098818569804721141489478423584171552557742706,

    69375809976016702099410374381169902756078062957317991387740995941852026]



       1/47 p[-33](47 n + 19), are all integers, the first, 5, terms are:

[84846389063172, 8757505280620040450363633965214832,

    1281621438661755271217248412574831248672388652938,

    616864681711984365243171730827682473217346584812112852069542,

    9696119806895729492476710640422758366543882903032791663833719462011565]



       1/59 p[-33](59 n + 53), are all integers, the first, 5, terms are:

[154890734937107626138180206213,

    547913371825561914823660356667615046350665721875,

    154522578128678787594996540675452999937118515903232569710146816,

    306134157865622236056051366409842593431616107775532743210839865698701815424

    , 24587294615683411792523736222897028304737338007993501495416370031013761\

    613553813764078]

Fact Number, 18, : There are no Congruences for, 35, colors with moduli up to ,

    200

                     --------------------------------------

                        Fact Number, 19, : There are, 3

              Congruences for, 37, colors modulo primes up to, 200



       1/41 p[-37](41 n + 34), are all integers, the first, 5, terms are:

[38494398953295642271905, 663315246729122570364114498248501846297,

    2733644791872427033109164494371466009171125002086014,

    147859266831654474494467723550603778382133274486097831066343495,

    490160197873691296210735448422773906041490862407761190876782410707775855]



       1/43 p[-37](43 n + 32), are all integers, the first, 5, terms are:

[4005425324753904405140, 632463374788233148486713823911362225539,

    9456686362641447936975017562937106099442448542591365,

    1331230749356335461338911233928775829202932080255903923224633323,

    9640506807714002459830849656126762102421650495553652918617914626474713565]



       1/47 p[-37](47 n + 27), are all integers, the first, 5, terms are:

[11117247280869696555, 263642452937358053992209973651680069840,

    58715147322755711848058185797636148695313556639264145,

    58338891126501385983702021353088377972807806739125682176320276190, 204209\

    1815886323773190714558751174635503965037822712100558945917904425421131]

                     --------------------------------------

                        Fact Number, 20, : There are, 3

              Congruences for, 39, colors modulo primes up to, 200



       1/47 p[-39](47 n + 31), are all integers, the first, 5, terms are:

[3361118186897513815227, 45846862979961176868094204312849602617175,

    12268092813777020452265611978203009519313940787347641323,

    16874054012727032146455449502463996610363059160730945833781060742350, 850\
    628856686443457607914694829879237715136712777308703355566036582890933807\

    795]



       1/53 p[-39](53 n + 48), are all integers, the first, 5, terms are:

[189165541338925157692416855670,

    1102241421298031820585792382075609160855149295367,

    534984035757411659195113415147674135881627472159589846450238360, 17947188\

    51379484324655516472757139792173832608073892373918022900677806535582, 238\
    379754093405518302840355004854596327965542263692440413311835113321079671\

    706456816644]



       1/61 p[-39](61 n + 55), are all integers, the first, 5, terms are:

[120634636484330963159247226857531,

    26868665408565850206690778960840642407675495037575519,

    204739871885370932077088069749492885318310493532337300802476052656283, 69\
    036805821650662440118244171513762812752323701477377931888408671160528043\

    91628845, 695841449978245478808324714679470030128055090943873269893086778\

    5055718024925850228981839950105]

                     --------------------------------------

                        Fact Number, 21, : There are, 1

              Congruences for, 41, colors modulo primes up to, 200



       1/47 p[-41](47 n + 35), are all integers, the first, 5, terms are:

[936494813354672245991933, 7983041420338024023911060451912580238555930,

    2529173823494485744017204865239154258654314100663754024120,

    4713653664557624990093170755902478042184672164861777177528183802389494, 3\
    351548659415971139922601463357845926471183090572403151317585277773811607\

    22971737]

                     --------------------------------------

                        Fact Number, 22, : There are, 1

              Congruences for, 43, colors modulo primes up to, 200



       1/47 p[-43](47 n + 39), are all integers, the first, 5, terms are:

[244798605024118394202959291, 1391676049971900957751351272956342206807184218,

    515285886341202925544951782232571855040700064906523899732971,

    1276386180142140449682660804415541037369431000684137269445320045570620701,
    125602741824028961641230062876512454391720352996662663696082023505220344\

    828489398280]

                     --------------------------------------

                        Fact Number, 23, : There are, 3

              Congruences for, 45, colors modulo primes up to, 200



       1/53 p[-45](53 n + 35), are all integers, the first, 5, terms are:

[6643977595959691052972343, 44018167878959906458898659496279708314134332850,

    1251488229612897881911827293562072634618371535146778856087443065, 8731575\

    2067096283236738691333004206240612766963119023349079363895552573558090, 1\
    403920326612293075836004516682547397354859094653503371174533329868245961\

    14590927879921125]



       1/59 p[-45](59 n + 24), are all integers, the first, 5, terms are:

[5633196419152309770, 696187787383757012349821384794859516773004400,

    2147117142425638462170384765659816351026506625141322219081726925, 3864165\

    923349832524697546077925797382307470247406117751467547553380873730576235,
    804017454581357336881433525197249500034121525588917771276405579793687471\

    38396695319833934071]



       1/71 p[-45](71 n + 64), are all integers, the first, 5, terms are:

[43195148165777519596973464668081178035,

    18423065394749695205543036908896496700152197580334011102377906, 500312657\

    57308313938674947803048211906333715867825711985286130181710948469297845,
    249293016444253801530632876319917718007414655020202671256963385273867060\

    346054671574608870283650, 21037287803597314161307901971221399336061061529\

    190506455225989240841687859418779038647970387851853861792083575]

Fact Number, 24, : There are no Congruences for, 47, colors with moduli up to ,

    200

                          This took, 1.937, seconds.