Problems & Puzzles: Puzzles

 Puzzle 1029. p that divides the number of partitions of p On Dec. 21, 2020, G.L. Honaker, Jr. made me notice this interesting prime curio from Matin Sariyar: The largest known prime p that divides P(p), the number of partitions of p. Note that P(5273) = 21324454654025779150078250115620526414565357199214419652174475578296382696055 = 5 × 7 × 17 × 37 × 5273 × 305971 × 182248 634924 894453 × 283809 003678 349559 × 11607 287812 587037 273754 619257. [Sariyar] Q. Can you find another example?

During the week Jan 30 to Feb 5, contributions came from Oscar Volpatti, Simon Cavegn.

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Oscar wrote:

I identified the first 17 numbers n that divide their partition number P(n):
1, 23, 124, 158, 342, 693, 1896, 3853, 4434, 5273, 8640, 14850, 17928, 110516, 178984, 274534.
Primes are marked in red. I checked all integers up to 2*10^7 and all primes up to 1.4*10^8, finding no more solutions.

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Simon wrote:

Found no larger such prime p. Searched up to p=32001516 where P(p) has 6294 digits, where I had to stop because the caching of subresults used more memory than the available 64GB.

Detailed results:
composite 124 = 2*2*31 divides 2841940500 = 2*2*3*5*5*5*7*31*8731
composite 158 = 2*79 divides 88751778802 = 2*79*691*853*953
composite 342 = 2*3*3*19 divides 164637479165761044 = 2*2*3*3*19*8761*27473812231
composite 693 = 3*3*7*11 divides 43397921522754943172592795 = 3*3*5*7*11*4297*2914743466632476779
composite 1896 = 2*2*2*3*79 divides 242371131052313431017875037233367567350390976 = 2*2*2*2*2*2*3*79*97*
1433*114956794400392328162963993391765607
prime 3853 divides 52458768232245476848382802651615548511564556404916972885720802646 = 2*3*19*
3853*3730147*514450134047737*62236476421501700408730868071603733517
composite 4434 = 2*3*739 divides 4893455653239535534170769252482853830477107691316358584295488910563020 =
2*2*3*5*53*97*739*1195673*69559095743*32540811588997133*7931907372622054599668315609
prime 5273 divides 21324454654025779150078250115620526414565357199214419652174475578296382696055 =
5*7*17*37*5273*305971*182248634924894453*283809003678349559*11607287812587037273754619257
composite 8640 = 2*2*2*2*2*2*3*3*3*5 divides 58840651254719691045710058970327680205626418675593060
6481023678358292584565247694401969643021683840 = 2*2*2*2*2*2*2*3*3*3*3*5*131*831907682800781533*
7566364064365160017*13765082328751949687977771035156301163986639016701117411
composite 14850 = 2*3*3*3*5*5*11 divides 549294505498275257800930826211910660370573352459330590038898
92895255992227974193459162563124100083276487430386006457751689030703150 = 2*3*3*3*3*5*5*7*11*43*173*
1010431*1166782481036437*178507590513453689*3592583976864797998651*313173329607563152636288263623
11704008128032645363805703100277
composite 17928 = 2*2*2*3*3*3*83 divides 1161798109138950759097487682256519723853483810051235075537307
302489424138594505837218771948173335239762460578353977112343398791066738837020305288 = 2*2*2*3*3*3*
83*27198614939*47800899562269173003*49844353054928249452503850834212360653850959547768625871940568
773801879335465501668601018887263256867375805513(composite)
composite 110516 = 2*2*7*3947 divides 2856346040839797463043936929459703449299174017813049579111148318
7948446894010279768905190678964988361330339612883922075551203585447519478403375443189832006018959
9367916029079704728816534178404420604164802922260813822735267000501429148377631615064546314054388
7194639469449378472150592518327912601410917854909589166802530004839992632150207775886276612943839
2623101008 = 2*2*2*2*3*3*7*13*3947*19512007481*93630721549*10251905296076473*294859347226387491260
4472854176747540252710029401786712940642021031650693243735538663200169203787401871539906772169293
3521334004477778310576084978973344022311162592049002649526458068086539188942924415722353359147531
0015730109667941504696774861726384023308457902645178632182507072691658462325029372460827759045733
13795993(composite)
composite 178984 = 2*2*2*13*1721 divides 1600213192811302323308433622476613891333064411563604975869204
6700609938861275534002329681961770347111622019199032104944900249728646467852048017777539491423969
5993000405879880352193170140130255116316401334177397506455083615402339446878058783363871168183658
3120589453733373333380842330282579179635870162733801000252989718074452593232777469616366979685854
7462698395322781957017663489777991868151367202494968119130191988758305511200433303674238922354410
18614156386448000 = 2*2*2*2*2*2*2*5*5*5*13*31*1721*14420222034725957873585687443647998553601695263
8369277184373001268539581671704066558568597605501835374201939829475701422403676095813104325490418
4762189179639196376180010682882739141668998799091758034249720081282328129200370066773376013235129
3598343642147352877365265784915477945427544817298317130799245470499798260843986725544338012349541
5734311418524231600972039695582422015587352716983426085366456390480381109863025766093988581726747
193828483807619009931(composite)
composite 274534 = 2*13*10559 divides 260789287175609545450661023948751995900416562902992493562366052329
135867140945623503018531952778705114364206987239122897666688312389372121511276788138428070303192389
242098764983553427997127074399523464545023162172574061892246225373450020882101487065161304423634220
915668543597320958369257544866015807474131921110404383064674989479346686587948337612126572803913594
395749043284918231279751499422357893004477872764702851099042434449248591079537734154012417811796481
610361279728265241914853634928247656189494941807285030907796694626274051182501669409616124022067748
09085237075205370 = 2*5*13*43*10559*4418299477382638681101405052362760606945055187860706261695142302
518819918961969102535332717763576115100822975749335286257875092056871883560680276448809035900381422
477126123150308377578803338365672392076230709801057793526014674598248601238735058817896428712326803
269941377932386753271535436452332770024573258619729642214259986427391239151603339978767006159001843
367149323196210512685763012187965668894223935202242969542791284452237094724735881981729588249615652
834975009317747320862445023286449336409375903824294987665488405547197425136692547920266344339305247
20182669577(composite)

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