Problems & Puzzles: Puzzles

Puzzle 341. Multiplicative persistence, Erdos style.

The original 'Multiplicative Persistence' concept is:

ďMultiply all the digits of a number N  by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence...

... The smallest numbers having multiplicative persistence of 1, 2, ... are 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (Sloane's A003001; Wells 1986, p. 78)....

There is no number N<10^50 with multiplicative persistence >11 (Wells 1986, p. 78)Ö.Ē

(Excerpt from Eric W. Weisstein. "Multiplicative Persistence." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MultiplicativePersistence.html )

This is the sequence of numbers arising from 277777788888899 (*):

277777788888899,  4996238671872, 438939648, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 0

2) Multiplicative Persistence, Erdos style, goes this way:

ďMultiply all the non-zero digits of a number N  by each other, repeating with the product until a single digit is obtained...."

Walter Schnider says that "Using the modification from Erdos the persistence seems not to be bounded (proof?)"

I have obtained the following n values for persistence greater than 11:

persistence  n (smallest?) value
12 (5)16 (7)13
13 (7)42(8)2(9)14
14 (2)1(6)1(7)130(9)8

This is the sequences for persistence 12:

55555555555555557777777777777, 14784089722747802734375, 49962386718720, 438939648, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 2
 

Questions:

1. The n values that I have obtained for persistence 12, 13 & 14 are the smallest values?

2. Can you obtain the smallest n values for persistence from 15 to 20?

3. Is the persistence (Erdos style) unbounded, as Schnider thinks?

_____
(*) BTW,
277777788888899 is the minimized expression of the number really computed originally, that is to say 22222222222222222223333777777. I have computed only one more distinct number composed only of digits 2, 3, & 7 having persistence 11: 22223333333333333333333377777, whose minimized expression is 27777789999999999. I bet that these two numbers (and all its permutations) are the only solutions - for numbers composed only of digits 2, 3 & 7 - for persistence 11.

 


Contributions came from Wilfred Whiteside & Phil Carmody:

***

WW wrote:

Puzzle 341 - minimal solutions to persistence 12 through 16

(Question1: case 12,13,14 solutions below are smaller than those posted by CR)

case of persist=12: (7)9 (8)12 (9)5 = 26 digits = minimum
77777777788888888888899999
163747527548676077518848
4996238671872000
438939648
4478976
338688
27648
2688
768
336
54
20
2

case of persist=13: (3)1 (7)27 (8)6 (9)19 = 53 digits = minimum
37777777777777777777777777778888889999999999999999999
69809726478979191696277299862872985495399563264
1404492781386126176471740317696000
24975483372306432
4389396480
4478976
338688
27648
2688
768
336
54
20
2

case of persist=14: (2)1 (6)1 (7)1 (8)99 (9)10 = 112 digits = minimum
2678888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999
74578405298868358061499745611046025877675420408378594099674468879366865988719527897725397266121883648
797466391371269339802853647433701338561899365731842129920000000000
1404492781386126176471740317696000
24975483372306432
4389396480
4478976
338688
27648
2688
768
336
54
20
2

Question2: In Progress

case of persist=15: (6)1 (7)157 (8)46 (9)25 = 229 digits = minimum
6777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777788888888888888888888888888888888888888888888889999999999999999999999999
7190277805656896237266279928179772059796081093791892500479162183359964757353956754361577464590374547304773629978879224056047498957910439999134424123798313948633871716972951634608296401150495846039552
6761649844889427899714485661836135982253298692044782055127319048849899874922975246243864709837052968960000000000000000000
478171229990877345192708930286667593551490390339639585013760000000
76808198982053775275798298624000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

case of persist=16: (3)1 (7)54 (8)82 (9)353 = 490 digits = minimum
3777777777777777777777777777777777777777777777777777777888888888888888888888888888888888888888888888888888888888888888888888888888888888899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
10313360079197038826077089364230457379432574159540187898047551374783782810031383346839773167884842406424617596932068662374149873895560771196842434034106623844332439337889427880983704625242093937990922999271273650634388894340185325262986279807209440122274073367109798977182804436148033972294151277222034341895648455313751129874315672779020852685493964028556348278697469898081571840854299775963529025551901462407830348077712377075714821070019280962881566277632
1976933552197413612159628902627430978010851694042125816551551749164417593934812396670920758208266066117717375273336106396150759715023578737998568950408664664249561214198331979010971731603753185698045337798713099904267181475705651200000000000000000000000000000000
6761649844889427899714485661836135982253298692044782055127319048849899874922975246243864709837052968960000000000000000000000000
478171229990877345192708930286667593551490390339639585013760000000
76808198982053775275798298624000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2
 

case of persist=17: (3)1 (7)27 (8)622 (9)399 = 1049 digits = minimum
37777777777777777777777777778888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
574779817804390904890286981578964902781742714480320102678593558479177180582606355263905759684722351577904879247007053974584208744001683336359195814522257396582927450017759678139720092317593733121971548881593674829563639377365298522834542493377570399998139939997206867358132738209223116041773016336986393741009927170577275675516663119149156228199482628326795805605312859951078492366250880322089839944278485593967559483726298470729579638656671875978230497021973201423578244054142219544517517323730416132555518599958484932098350355976621826733775403353089175239106643656216106760961306951699339732730932329296161823783063407785293611090217664324462726977118668779311130064512708621524267150762503281355753486157669279028423101647557295097183503597630687186194628637661562365592379177125730665243432740056266434025929842174673602217195744165721547122876094729663369401715352412452489940215312319347759504339884598957640155052631453987231074795542859487259944471949737984
828368498364837666825200579884512979791181474317446829688303086334406913900460872056195260292948438666866245620375776835468872736609243559801402574861042255424847483347504475870445862265647578973527779685846893028443330223238867730328672262060722901509882728834544281291989982007388436910606023096673175842971489593137979851445568782398280193277780127646514774127076336897004902895175816443798802252380437662173047599682950648247296300971464012595200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1047621388678982390523975179652349599477748169802451387943572170708460577790233282360940599731586471226145975582852646268709258217365333296402680064553776311480647684769193600633649249762058762715383618293106618568275937262672281600000000000000000000000000000000000
1027981992925234360360281305873470377951167677089379366226989667956442830719965534999107611234813238663498319265792000000000000000000000
69040999053800788558896662715231492793697575191617305376194560000000
7680819898205377527579829862400000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

Since number of digits approx. doubles for each increase of persistence ... will the next one be about 2200 digits?

...

I just found that Sloane lists up to N=13 for puzzle 341.
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A014120

***

Phil wrote:

The Mathworld article you quote is out of date. See
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0107&L=NMBRTHRY&P=R1036&I=-3

***

case of persist=18: (3)1 (7)140 (8)258 (9)1946 = 2345 digits = minimum
3777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777778888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
5546089446864993529211309277721902623921032942997752811155459225454653157758875848216708167539884748935256311209720493184916224374011538798648276693622901916484264375777973990118071990405792714953864728927670012650548636232907369331755715064164141721771477290463405073729397681549209707582251480573313202793384449720114125873722456567562884760557716994630379279118028796491275961180600131863361999229306561907902495863924270980338162667096065910767199498570847066832858969748012938702706242895453715720942053399871597306066885767178386889651213451677381892954610218189664453784722038903379574718002867001824430885557853947001735683243880129729029326901329441072385714478176524063707618271395123427178392244597065078280601777333294867808651451993623108209418834504133607836522285959938918017616536647743660794464010778591673043817853791825738359957977731732339111065870691499407301026683534386928847776717097640892983356648443269088620338566007602443823504685441911392453914455515814960267177013710458463913662218117868875394356288738109936930025612349179764792739182147743766499893101451159488613401271011428659470767788899767225322028665874820423656707171148149345649701223424038606655403674471053129707799443096383054475131893524143060453492257965492908867560190324635167174136171969783733686211350810179264527500393801257803155996718586021917228213197087852008170483234719413171993921675905733555859244024039834888419510755002530218965368674915121610588239589313431121133377748320919368045315707926392390573792935382942764152735873140708200903102871357107928179207372569882789469200172668691784641049939859933638688576573068442782610447792967163415984420992534012210694727327402017165902469331478723572913714337604928465150355702762005047183733799690728706783422518348196433756560679980996138369586377726077245285392106166028866336175476921565145323228832943092270880811262867881292120461534401133519772059778555272625658599275254701443447484193914491970829305644077317506854180148207593585397487833743271651902684680051069928953143296132134878
004081045658853357675509678872299543196936893471271548167163098933196555956991970543615801477763277391640108271269167995416226338481922874767886525174355362578432
467338002977719346258635144194285880902335387146283511718071933474044456888783142691112725858124329244990441918382247336664254516542386750620984965587975475569799829106885984759652281988913364369724740961798915637708611954432931159979158114951095682019026006507502163256855675727249077525851367064446411174638256732956862893695340701395932509242548101702336832171179906929978279089043353918445104687806701933961692810499705360223461538019631997602791280161641012393929504868786066660127991769859891806683048666885286312636697448189392773404726062086934371836289043776441956148930392693334789998938242867177309932777596083665170729974555087375022873915131740301901340080187317590584808459367963873486550406189099383135020761554472032741278749942096060900983773251658639447264177829960064934488350906777019936284789806104263475489585429876692932887656975405614281052659569663873471683378070950996019224571716010153807508395660676699836493866257770779183016238093468466275300297683143931779399818379871564965579206760474245623899389300143337714483200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5282528022771627962616697821876118575425232316103621107312801043126357684026493696376850636823916301534209864376562322666511263786106379015698812556929987122537856402332615807203303209820558200812088431298970866764261440361930929766254975271956297170432736665516954279982826687403285241255658656754484389619293415109709835696321181437683470340536773600363074739226159462530122576218292298490399124152530248701689677889813140153335392203459989622290032378832683687838134781221439849041952310952292407276430483523892997521408000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
67185664822176592087211503596296259026338640541054584459171760071743803304205757654780034195294194593728172227168327493651763035191142648597386774481837004832375464349798584529195748969608198380851909577961747885909503301738862534678267559936000000000000000000000000000000000000000000000
10279819929252343603602813058734703779511676770893793662269896679564428307199655349991076112348132386634983192657920000000000000000000000000000
69040999053800788558896662715231492793697575191617305376194560000000
7680819898205377527579829862400000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

***

Marc Lapierre wrote on August 12, 2013:

To complete your list, here are some results with numbers having greater persistence than 18 :
p( (5)1(8)6457 ) = 19
p( (8)14829 ) = 20
p( (8)32935 ) = 21
p( (8)68764 ) = 22
Please note that these solutions are not minimal.
They can be tested here :
or better with your own tools to be sure there is no mistake :)
It really seems to be unbounded when you look at the shape of the curve in above url...
I heard about this problem thanks to the last paper of JP Delahaye author of mathematics pages in Pour la Science (french version of Scientific American).

***

 

On Set 23, 2017, Edouard Debonneuil wrote:

I found your page http://www.primepuzzles.net/puzzles/puzz_341.htm as a person at work told me about the persistence-limit
 conjecture and I found the answer; I am going to make an article, but your page indicates to email you to add a solution; could you add this? :


a) I looked for persistence greater than 11 up to 10^1000 and found none. It took a few few hours, so I am planning to parallelize
 it to go to 10^10000.Method: I cycled through 2^i*3^j*7*k and 3^i*5^j*7^k up to that limit (limit: eg 237788999 is written
 in less than 1000 digits). The known persistance 11 numbers are found quasi immediately and then no other persistance 11 number appears.


b) I found why persistance is limited: the number of possible "links" (to go down from one number to a single digit, when computing
 the persistence of a number) that don't have a "0" is limited (so it is impossible to build a longer chain of links, ie a greater
 persistence, than that limit):


-- these links are written as 2^i*3^j*7*k and 3^i*5^j*7^k (otherwise they have a zero at the end, when written)
-- there are asymptotically O(n^2) numbers "2^i*3^j*7*k and 3^i*5^j*7^k" of size n [proof (*)]; what counts is that it is polynomial
-- an asymptotic proportion A*exp(-B n) of them don't have zeros (if "2^i*3^j*7*k and 3^i*5^j*7^k" have as many "0" as other
 numbers of the same size without a 0 at the end, the proportion is (9/10)^(n-1)
-- summing over all lengths n (to sum over all integers), the integral of a polynomial times an exponential is defined (not infinite)


c) The reasoning is valid of other fixed numbering bases > 3


d) For persistance a la Erdos, the same type of reasoning suggests(**) that there is no limit:
-- There are O(n^3) possible links of size n [2^i*3^j*5^k*7^l]
-- (**) Assuming that the product of their digits behaves like other numbers of the same size, there are O(n^a) links-of-links of size n, etc.
-- So whatever the desired persistence p there is an infinite number of chains that get to that persistence, a la Erdos.


(*) Considering 2^i*3^j*7*k numbers,
10^n <= 2^i*3^j*7*k < 10^(n+1) can be written n <= i*log(2)+j*log(3)+k*log(7) < n+1
 i*log(2)+j*log(3)+k*log(7) < K can be visualized as the volume of half a cube whose connected tips are (0,0,0), (K/log(2),0,0),
 (0,K/log(3),0), (0,0,K/log(7)) and the cube is cut in half by the plane that goes through  these 3 latter tips.
 So asymptotically the volume is K^3/(log2*log3*log7).

We want the tranche from K-1 to K (n to n+1) so we derive by K: the volume of the tranche is 3*K^2/(log2*log3*log7)
So card({2^i*3^j*7*k or 3^i*5^j*7*k | size n) ~ 3*n^2*(1/log2+1/log5)/(log3*log7)

***

On August 5, 2020, Julien Courties and Mael Courtain wrote:
We (MaŽl Courtain & Julien Courties) found a solution for persistence 26.
The number is (8)8915882,  known by a random approach (on Python program).
The numbers obtained for each iteration are too large to be written out entirely.
Therefore, we wrote them by constitutions of ni digits, where i varies from 0 to 9.
Thanks to Jean-Paul DELAHAYE and Marc LAPIERRE for their help and advice.

Please find below the 26 iterations result :

n2=0 n3=0 n4=0 n5=0 n6=0 n7=0 n8=8915882 n9=0
n0 = 804340 n1 = 805587 n2 = 805197 n3 = 804733 n4 = 806043 n5 = 805511 n6 = 804999 n7 = 803602 n8 = 806452 n9 = 805380
n0 = 1172899 n1 = 367206 n2 = 365717 n3 = 366696 n4 = 366167 n5 = 368421 n6 = 367573 n7 = 367620 n8 = 367035 n9 = 367684
n0 = 536259 n1 = 167229 n2 = 167162 n3 = 167502 n4 = 167568 n5 = 167866 n6 = 167337 n7 = 167411 n8 = 166880 n9 = 166834
n0 = 243492 n1 = 75968 n2 = 76500 n3 = 76306 n4 = 76005 n5 = 76357 n6 = 76533 n7 = 76160 n8 = 76152 n9 = 76586
n0 = 111240 n1 = 34703 n2 = 34891 n3 = 34526 n4 = 34508 n5 = 35019 n6 = 34558 n7 = 34830 n8 = 35079 n9 = 34984
n0 = 50934 n1 = 15777 n2 = 15932 n3 = 15776 n4 = 15879 n5 = 15911 n6 = 15951 n7 = 15843 n8 = 15779 n9 = 15837
n0 = 23148 n1 = 7045 n2 = 7092 n3 = 7309 n4 = 7148 n5 = 7285 n6 = 7228 n7 = 7329 n8 = 7265 n9 = 7319
n0 = 10587 n1 = 3364 n2 = 3321 n3 = 3296 n4 = 3393 n5 = 3310 n6 = 3328 n7 = 3237 n8 = 3327 n9 = 3218
n0 = 4755 n1 = 1491 n2 = 1506 n3 = 1473 n4 = 1539 n5 = 1507 n6 = 1545 n7 = 1479 n8 = 1475 n9 = 1560
n0 = 2186 n1 = 690 n2 = 698 n3 = 709 n4 = 660 n5 = 679 n6 = 655 n7 = 719 n8 = 707 n9 = 706
n0 = 1009 n1 = 324 n2 = 321 n3 = 293 n4 = 323 n5 = 324 n6 = 342 n7 = 327 n8 = 305 n9 = 282
n0 = 463 n1 = 130 n2 = 177 n3 = 149 n4 = 145 n5 = 138 n6 = 143 n7 = 136 n8 = 141 n9 = 123
n0 = 205 n1 = 81 n2 = 71 n3 = 55 n4 = 59 n5 = 68 n6 = 62 n7 = 59 n8 = 66 n9 = 54
n0 = 92 n1 = 26 n2 = 27 n3 = 30 n4 = 28 n5 = 28 n6 = 31 n7 = 19 n8 = 30 n9 = 29
n0 = 44 n1 = 11 n2 = 20 n3 = 11 n4 = 11 n5 = 17 n6 = 10 n7 = 17 n8 = 5 n9 = 8
n0 = 18 n1 = 6 n2 = 2 n3 = 6 n4 = 9 n5 = 4 n6 = 9 n7 = 5 n8 = 2 n9 = 4
n0 = 6 n1 = 0 n2 = 2 n3 = 3 n4 = 2 n5 = 0 n6 = 3 n7 = 4 n8 = 4 n9 = 5
n0 = 1 n1 = 3 n2 = 2 n3 = 1 n4 = 1 n5 = 3 n6 = 3 n7 = 1 n8 = 0 n9 = 3
n0 = 3 n1 = 1 n2 = 0 n3 = 1 n4 = 1 n5 = 0 n6 = 2 n7 = 0 n8 = 2 n9 = 0
n0 = 0 n1 = 0 n2 = 1 n3 = 0 n4 = 1 n5 = 0 n6 = 1 n7 = 1 n8 = 1 n9 = 0
n0 = 0 n1 = 0 n2 = 1 n3 = 0 n4 = 0 n5 = 0 n6 = 1 n7 = 0 n8 = 2 n9 = 0
n0 = 0 n1 = 0 n2 = 0 n3 = 0 n4 = 0 n5 = 0 n6 = 1 n7 = 1 n8 = 1 n9 = 0
n0 = 0 n1 = 0 n2 = 0 n3 = 2 n4 = 0 n5 = 0 n6 = 1 n7 = 0 n8 = 0 n9 = 0
n0 = 0 n1 = 0 n2 = 0 n3 = 0 n4 = 1 n5 = 1 n6 = 0 n7 = 0 n8 = 0 n9 = 0
n0 = 1 n1 = 0 n2 = 1 n3 = 0 n4 = 0 n5 = 0 n6 = 0 n7 = 0 n8 = 0 n9 = 0
n0 = 0 n1 = 0 n2 = 1 n3 = 0 n4 = 0 n5 = 0 n6 = 0 n7 = 0 n8 = 0 n9 = 0
 
Python code (that takes about 12h to run on a personal computer) :
def main():
    n=20
    n2=0
    n3=0
    n4=0
    n5=0
    n6=0
    n7=0
    n8=8915882
    n9=0
    cpt = 0
    while (n >= 10):
        n = 1
        for i in range(0, n2):
            n = n * 2
        for i in range(0, n3):
            n = n * 3
        for i in range(0, n4):
            n = n * 4
        for i in range(0, n5):
            n = n * 5
        for i in range(0, n6):
            n = n * 6
        for i in range(0, n7):
            n = n * 7
        for i in range(0, n8):
            n = n * 8
        for i in range(0, n9):
            n = n * 9
        l = len(str(n))
        decompo = (str(n))
        n0 = 0
        n1 = 0
        n2 = 0
        n3 = 0
        n4 = 0
        n5 = 0
        n6 = 0
        n7 = 0
        n8 = 0
        n9 = 0
        for i in range(0, l):
            if (decompo[i] == str(2)):
                n2 = n2 + 1
            if (decompo[i] == str(3)):
                n3 = n3 + 1
            if (decompo[i] == str(4)):
                n4 = n4 + 1
            if (decompo[i] == str(5)):
                n5 = n5 + 1
            if (decompo[i] == str(6)):
                n6 = n6 + 1
            if (decompo[i] == str(7)):
                n7 = n7 + 1
            if (decompo[i] == str(8)):
                n8 = n8 + 1
            if (decompo[i] == str(9)):
                n9 = n9 + 1
            if (decompo[i] == str(0)):
                n0 = n0 + 1
            if (decompo[i] == str(1)):
                n1 = n1 + 1
        cpt = cpt + 1
        print("n0 = ", n0, " n1 = ", n1, " n2 = ", n2, " n3 = ", n3, " n4 = ", n4, " n5 = ", n5, " n6  ", n6," n7 = ", n7, " n8 = ", n8, " n9 = ", n9)
    print("iterations = ", cpt)


if __name__ == '__main__':
    main()

Does anybody has time and patience enough to confirm this result?

***

On August 27, MaŽl Courtain & Julien Courties wrote again:

We (always MaŽl Courtain & Julien Courties) found a new record for the Erdos multiplicative persistence. 
Indeed, this time, the number of iterations is 27 with (7)5032475.

 
Please find below the the full list of calculation results :

 
n2 =  0 n3 =  0  n4 =  0  n5 =  0  n6 =  0  n7 =  5032475  n8 =  0  n9 =  0
n0 =  425184  n1 =  425252 n2 =  425140  n3 =  424991  n4 =  424628  n5 =  425814  n6 =  425401  n7 =  425784  n8 =  424799  n9 =  425942
n0 =  619660  n1 =  194377 n2 =  193473  n3 =  193641  n4 =  193942  n5 =  194177  n6 =  193949  n7 =  193521  n8 =  193713  n9 =  194522
n0 =  282311  n1 =  88361 n2 =  87829  n3 =  88634  n4 =  88557  n5 =  88134  n6 =  88043  n7 =  88777  n8 =  88742  n9 =  88760
n0 =  128435  n1 =  40669 n2 =  40010  n3 =  40442  n4 =  40288  n5 =  40429  n6 =  40315  n7 =  40538  n8 =  40330  n9 =  40569
n0 =  58930  n1 =  18095 n2 =  18619  n3 =  18469  n4 =  18304  n5 =  18288  n6 =  18551  n7 =  18542  n8 =  18346  n9 =  18475
n0 =  26711  n1 =  8435 n2 =  8446  n3 =  8582  n4 =  8373  n5 =  8548  n6 =  8390  n7 =  8401  n8 =  8338  n9 =  8299
n0 =  12317  n1 =  3744 n2 =  3825  n3 =  3808  n4 =  3888  n5 =  3827  n6 =  3857  n7 =  3792  n8 =  3826  n9 =  3847
n0 =  5564  n1 =  1682 n2 =  1756  n3 =  1831  n4 =  1749  n5 =  1765  n6 =  1719  n7 =  1687  n8 =  1786  n9 =  1778
n0 =  2539  n1 =  799 n2 =  831  n3 =  827  n4 =  776  n5 =  739  n6 =  807  n7 =  821  n8 =  802  n9 =  821
n0 =  1092  n1 =  368 n2 =  322  n3 =  396  n4 =  376  n5 =  374  n6 =  355  n7 =  385  n8 =  400  n9 =  390
n0 =  530  n1 =  167 n2 =  158  n3 =  181  n4 =  182  n5 =  186  n6 =  201  n7 =  174  n8 =  167  n9 =  163
n0 =  267  n1 =  83 n2 =  87  n3 =  83  n4 =  84  n5 =  78  n6 =  88  n7 =  65  n8 =  73  n9 =  76
n0 =  112  n1 =  37 n2 =  32  n3 =  40  n4 =  44  n5 =  36  n6 =  28  n7 =  36  n8 =  34  n9 =  34
n0 =  46  n1 =  18 n2 =  17  n3 =  14  n4 =  20  n5 =  22  n6 =  21  n7 =  4  n8 =  15  n9 =  19
n0 =  32  n1 =  5 n2 =  4  n3 =  8  n4 =  5  n5 =  11  n6 =  6  n7 =  1  n8 =  11  n9 =  8
n0 =  12  n1 =  1 n2 =  5  n3 =  6  n4 =  2  n5 =  0  n6 =  5  n7 =  5  n8 =  3  n9 =  0
n0 =  1  n1 =  0 n2 =  3  n3 =  4  n4 =  3  n5 =  1  n6 =  1  n7 =  2  n8 =  1  n9 =  1
n0 =  1  n1 =  0 n2 =  0  n3 =  2  n4 =  2  n5 =  0  n6 =  1  n7 =  0  n8 =  2  n9 =  2
n0 =  0  n1 =  0 n2 =  0  n3 =  0  n4 =  2  n5 =  0  n6 =  1  n7 =  2  n8 =  1  n9 =  1
n0 =  0  n1 =  0 n2 =  0  n3 =  2  n4 =  0  n5 =  0  n6 =  1  n7 =  0  n8 =  3  n9 =  0
n0 =  0  n1 =  0 n2 =  1  n3 =  0  n4 =  1  n5 =  0  n6 =  1  n7 =  1  n8 =  1  n9 =  0
n0 =  0  n1 =  0 n2 =  1  n3 =  0  n4 =  0  n5 =  0  n6 =  1  n7 =  0  n8 =  2  n9 =  0
n0 =  0  n1 =  0 n2 =  0  n3 =  0  n4 =  0  n5 =  0  n6 =  1  n7 =  1  n8 =  1  n9 =  0
n0 =  0  n1 =  0 n2 =  0  n3 =  2  n4 =  0  n5 =  0  n6 =  1  n7 =  0  n8 =  0  n9 =  0
n0 =  0  n1 =  0 n2 =  0  n3 =  0  n4 =  1  n5 =  1  n6 =  0  n7 =  0  n8 =  0  n9 =  0
n0 =  1  n1 =  0 n2 =  1  n3 =  0  n4 =  0  n5 =  0  n6 =  0  n7 =  0  n8 =  0  n9 =  0
n0 =  0  n1 =  0 n2 =  1  n3 =  0  n4 =  0  n5 =  0  n6 =  0  n7 =  0  n8 =  0  n9 =  0

 
Also, we would like you to know that we were able to complete the data for 24 iterations by giving the smallest numbers we found for each digit,
and we are trying to find it for 25 iterations. So please find here the status of our research, and note that the blue is used to present the
results of Marc Lapierre's calculations, and the yellow to ours :

 

k = pE (digit)n digit 2 digit 3 digit 4 digit 5 digit 6 digit 7 digit 8 digit 9
23 430411 294544 223924 185486 175085 153749 149497 147272
24 929761 629784 489694 391728 348883 351738 314985 267376
25             800208  
26           3107869 8915882  
27           5032475    

***

 

 

 

 


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