# Sloane persistence problem in every base between 3 and 100

Tags: mathématiques

Let $f_b(n)$ be the product of the digits of $n$ written in base $b$, let $p_b(n)$ the the least integer $k$ such that $f_b^k(n)\lt b$ and let $p_b$ be the maximum value that $p_b$ can get on the integers. This is known as the Sloane persistence problem.

It is conjectured that $p_3 = 3$ and $p_{10} = 11$.

Here is lower bound I got (without hard effort) for $p_b$, for $3\le b\le 100$, with a number that gives the bound. (The value before $\times$ is easily factorizable into numbers less than $b$).

(!: the bound is the conjectured value (by me).)

$p_{3} \ge 3$ (!) with $1\times 222$

$p_{4} \ge 3$ (!) with $1\times 333$

$p_{5} \ge 6$ (!) with $1\times 3344444444444444444444$

$p_{6} \ge 5$ (!) with $2\times 5555$

$p_{7} \ge 8$ (!) with $2\times 3334444555555555555$

$p_{8} \ge 6$ (!) with $1\times 333555577$

$p_{9} \ge 7$ (!) with $2\times 577777$

$p_{10} \ge 11$ (!) with $2\times 77777788888899$

$p_{11} \ge 13$ (!) with $3\times 555555555555555555557777777777777777788888888888888999999$

$p_{12} \ge 7$ (!) with $3\times 577777799$

$p_{13} \ge 15$ (!) with $3\times 5555555577788888899999999999999999999999$

$p_{14} \ge 13$ (!) with $2\times 5555555899999999999999(11)(11)(11)(11)(11)(11)$

$p_{15} \ge 11$ (!) with $2\times 788899(11)(11)(11)(11)(13)(13)(13)(13)(13)(13)$

$p_{16} \ge 8$ (!) with $3\times 79(11)(13)(13)$

$p_{17} \ge 18$ with $4\times 55555555555579999999999999999999999(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)$

$p_{18} \ge 10$ (!) with $4\times 555555(16)(17)$

$p_{19} \ge 19$ with $12\times 55555577777777799999(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(13)(13)(13)(13)(13)(13)(16)(16)(16)(16)(16)(16)(16)(16)(16)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)$

$p_{20} \ge 14$ with $24\times 79999(13)(13)(13)(13)(16)(16)(16)(17)(17)(17)(17)(17)(17)$

$p_{21} \ge 16$ with $1\times 559(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(11)(13)(16)(16)(16)(16)(16)(16)(16)(19)(19)(19)(19)(19)(19)(19)(19)$

$p_{22} \ge 17$ with $12\times 55777779999(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(19)(19)$

$p_{23} \ge 21$ with $6\times 559999(11)(11)(13)(13)(13)(13)(13)(13)(13)(13)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(16)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(17)(19)$

$p_{24} \ge 10$ with $4\times 5(16)(16)(16)(19)(19)(19)$

$p_{25} \ge 15$ with $8\times 79(13)(13)(13)(13)(13)(16)(16)(16)(16)(19)(23)(23)(23)(23)$

$p_{26} \ge 20$ with $12\times 77777779(11)(16)(16)(16)(16)(16)(17)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(19)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)$

$p_{27} \ge 12$ with $1\times 77777(13)(17)(19)(25)(25)$

$p_{28} \ge 18$ with $2\times (11)(11)(11)(11)(11)(11)(11)(13)(13)(13)(16)(16)(16)(16)(16)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(25)(25)(25)(25)(25)(25)(25)$

$p_{29} \ge 24$ with $360\times 77(11)(11)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(13)(16)(16)(16)(16)(16)(16)(19)(19)(19)(19)(19)(19)(19)(19)(19)(23)(23)(23)(23)(23)(23)(25)(25)(25)(27)$

$p_{30} \ge 15$ with $9\times 777(17)(19)(19)(19)(19)(19)(19)(29)(29)(29)$

$p_{31} \ge 22$ with $6\times 777(11)(13)(13)(13)(13)(13)(17)(17)(17)(23)(25)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(27)(29)(29)(29)(29)(29)(29)(29)(29)(29)(29)(29)$

$p_{32} \ge 12$ with $45\times (11)(13)(13)(25)(27)$

$p_{33} \ge 19$ with $45\times 7777(23)(23)(23)(23)(23)(23)(23)(23)(25)(25)(25)(25)(25)(25)(27)(27)(27)(27)(27)(27)(29)(31)(31)(31)(31)(32)(32)(32)(32)$

$p_{34} \ge 21$ with $360\times (13)(13)(13)(23)(23)(25)(25)(27)(27)(27)(27)(27)(27)(27)(27)(29)(29)(29)(29)(31)(31)(31)(31)(32)(32)(32)(32)(32)(32)(32)$

$p_{35} \ge 18$ with $720\times (13)(13)(13)(13)(13)(17)(17)(23)(23)(23)(27)(27)(27)(27)(27)(29)(29)(31)(31)(31)(31)(31)(31)(32)(32)$

$p_{36} \ge 13$ with $40\times (11)(11)(13)(19)(19)(19)(23)(23)(23)(23)(25)$

$p_{37} \ge 23$ with $6\times (13)(13)(13)(13)(17)(23)(25)(25)(25)(29)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(32)(32)(36)$

$p_{38} \ge 22$ with $40\times (11)(17)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(23)(25)(32)(32)(32)(36)(36)(36)(36)(36)(36)(36)(37)(37)(37)(37)$

$p_{39} \ge 22$ with $48\times 77(23)(23)(23)(25)(25)(27)(29)(29)(29)(31)(31)(31)(31)(31)(36)(36)(36)(36)(37)(37)(37)(37)(37)(37)(37)(37)$

$p_{40} \ge 17$ with $24\times 7(13)(17)(17)(17)(17)(19)(19)(29)(29)(29)(29)(31)(31)(32)(36)$

$p_{41} \ge 24$ with $18\times (13)(17)(17)(17)(23)(23)(23)(23)(23)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(29)(29)(31)(36)(36)(36)(37)$

$p_{42} \ge 20$ with $6\times (13)(13)(17)(19)(19)(19)(19)(19)(23)(27)(27)(27)(27)(27)(29)(29)(29)(29)(29)(29)(31)(31)(31)(36)(36)(37)(41)$

$p_{43} \ge 23$ with $40\times (13)(13)(13)(23)(23)(25)(29)(29)(31)(32)(32)(32)(32)(32)(37)(37)(37)(41)(41)(41)(41)(41)(41)(41)$

$p_{44} \ge 21$ with $18\times (19)(25)(25)(25)(25)(25)(25)(25)(29)(29)(29)(41)(41)(41)(41)(41)(41)(41)(41)(41)(43)(43)$

$p_{45} \ge 17$ with $6\times 7777(13)(13)(13)(13)(13)(17)(17)(17)(17)(19)(19)(31)(31)(31)(31)(37)(37)(43)(43)$

$p_{46} \ge 23$ with $240\times (11)(19)(27)(27)(29)(29)(29)(29)(31)(37)(37)(41)(41)(41)(41)(41)(41)(41)(41)(43)(43)(43)(43)(43)$

$p_{47} \ge 23$ with $240\times (11)(13)(17)(23)(23)(23)(23)(23)(25)(27)(29)(29)(31)(31)(41)(41)(43)(43)(43)(43)(43)$

$p_{48} \ge 13$ with $20\times 77(11)(11)(11)(13)(25)(32)(41)(47)(47)$

$p_{49} \ge 21$ with $3\times (11)(17)(25)(29)(29)(29)(29)(29)(29)(32)(32)(32)(36)(36)(36)(36)(36)(37)(41)(41)(43)(47)(47)$

$p_{50} \ge 19$ with $84\times (11)(11)(11)(11)(11)(19)(29)(29)(29)(29)(36)(36)(36)(36)(49)(49)(49)(49)(49)$

$p_{51} \ge 22$ with $4\times (11)(11)(11)(19)(29)(29)(29)(29)(32)(32)(36)(47)(47)(47)(47)(47)(49)(49)(49)(49)$

$p_{52} \ge 23$ with $42\times (19)(19)(19)(19)(19)(23)(23)(29)(29)(29)(29)(31)(31)(31)(32)(32)(37)(37)(41)(41)(47)$

$p_{53} \ge 27$ with $84\times (17)(17)(23)(25)(32)(32)(32)(32)(32)(36)(37)(37)(37)(41)(47)(47)(49)(49)$

$p_{54} \ge 17$ with $8\times (13)(17)(23)(25)(25)(25)(25)(25)(29)(29)(32)(43)(49)$

$p_{55} \ge 21$ with $240\times (19)(23)(23)(23)(25)(25)(27)(36)(36)(41)(41)(47)(47)(47)(53)(53)$

$p_{56} \ge 19$ with $240\times (11)(11)(13)(25)(27)(29)(36)(37)(37)(37)(37)(37)(37)(41)(47)(47)(53)(53)$

$p_{57} \ge 24$ with $840\times (13)(17)(17)(17)(32)(37)(41)(43)(43)(43)(43)(47)(47)(47)(49)(49)(49)(49)(49)$

$p_{58} \ge 27$ with $210\times (17)(23)(31)(31)(32)(32)(36)(36)(36)(36)(36)(37)(41)(43)(43)(47)(47)(49)(53)$

$p_{59} \ge 26$ with $210\times (11)(11)(17)(19)(25)(32)(32)(32)(37)(41)(41)(41)(47)(47)(47)(49)(49)(53)$

$p_{60} \ge 18$ with $21\times (11)(13)(13)(13)(27)(31)(36)(36)(37)(37)(43)(43)(47)(49)(49)(49)(53)(53)$

$p_{61} \ge 26$ with $140\times (19)(29)(29)(32)(36)(41)(43)(53)(53)(59)(59)(59)(59)(59)(59)(59)(59)$

$p_{62} \ge 24$ with $240\times (23)(27)(36)(36)(41)(47)(47)(47)(47)(53)(53)(53)(53)(53)(59)(61)(61)$

$p_{63} \ge 20$ with $20\times (19)(19)(23)(23)(36)(36)(36)(41)(41)(41)(41)(41)(41)(59)(61)(61)$

$p_{64} \ge 15$ with $5\times (13)(23)(25)(27)(37)(37)(37)(53)$

$p_{65} \ge 24$ with $210\times (11)(17)(25)(25)(27)(36)(49)(49)(53)(53)(53)(61)(64)(64)(64)$

$p_{66} \ge 21$ with $48\times (19)(31)(31)(31)(31)(47)(49)(49)(49)(53)(59)(59)(59)(61)(64)(64)$

$p_{67} \ge 28$ with $18\times (19)(37)(41)(41)(41)(43)(53)(59)(59)(64)(64)(64)(64)(64)$

$p_{68} \ge 23$ with $48\times (23)(31)(36)(43)(47)(49)(49)(49)(49)(59)(59)(61)(61)(61)(61)(67)(67)$

$p_{69} \ge 25$ with $63\times (31)(36)(41)(41)(41)(41)(41)(41)(53)(59)(59)(59)(59)(59)(67)(67)(67)$

$p_{70} \ge 21$ with $45\times (17)(17)(19)(36)(37)(41)(41)(41)(43)(47)(47)(47)(53)(53)(53)(59)(64)$

$p_{71} \ge 25$ with $9\times (19)(19)(27)(27)(27)(29)(29)(41)(49)(49)(53)(59)(59)(61)$

$p_{72} \ge 16$ with $20\times (11)(13)(19)(25)(47)(47)(59)(61)(61)(61)$

$p_{73} \ge 27$ with $24\times (17)(25)(31)(43)(49)(53)(61)(61)(61)(61)(61)(64)(71)(71)(71)$

$p_{74} \ge 24$ with $1680\times (13)(17)(31)(31)(47)(59)(59)(59)(61)(61)(67)(73)(73)$

$p_{75} \ge 22$ with $16\times (19)(19)(23)(31)(31)(37)(37)(37)(47)(53)(61)(61)(73)(73)(73)(73)$

$p_{76} \ge 25$ with $630\times (31)(36)(36)(36)(36)(37)(41)(43)(49)(49)(59)(61)(61)(61)(64)$

$p_{77} \ge 22$ with $315\times (23)(23)(37)(53)(53)(53)(67)(71)(71)(73)(73)(73)(73)$

$p_{78} \ge 25$ with $96\times (23)(27)(36)(36)(47)(49)(49)(49)(49)(53)(59)(64)(71)(73)$

$p_{79} \ge 26$ with $420\times (23)(37)(43)(43)(47)(53)(59)(59)(59)(59)(61)(67)(71)$

$p_{80} \ge 20$ with $14\times (17)(19)(29)(36)(41)(43)(43)(59)(61)(61)(64)(64)(64)(71)(71)(79)$

$p_{81} \ge 19$ with $7\times (25)(25)(25)(37)(37)(37)(49)(49)(49)(49)(49)(71)(73)$

$p_{82} \ge 25$ with $16\times (37)(53)(53)(59)(59)(59)(59)(71)(71)(73)(73)(81)(81)(81)$

$p_{83} \ge 26$ with $336\times (29)(29)(41)(43)(47)(67)(67)(67)(73)(73)(73)(73)(79)$

$p_{84} \ge 20$ with $27\times (11)(11)(25)(43)(53)(53)(61)(64)(67)(71)(79)(79)(79)$

$p_{85} \ge 24$ with $28\times (25)(47)(49)(49)(67)(73)(73)(73)(81)(81)(81)(81)(81)$

$p_{86} \ge 25$ with $8\times (17)(29)(41)(47)(59)(64)(64)(71)(73)(73)(73)(79)(81)$

$p_{87} \ge 29$ with $160\times (31)(41)(43)(49)(49)(49)(61)(64)(67)(67)(67)(71)(81)(81)$

$p_{88} \ge 22$ with $20\times (36)(41)(43)(47)(67)(71)(71)(71)(73)(81)(81)(81)(81)(81)$

$p_{89} \ge 27$ with $30\times (25)(31)(47)(49)(49)(49)(61)(71)(73)(79)(83)(83)$

$p_{90} \ge 21$ with $672\times (17)(31)(31)(31)(53)(59)(67)(67)(67)(73)(81)(81)(81)$

$p_{91} \ge 25$ with $630\times (11)(23)(23)(29)(37)(41)(41)(43)(47)(64)(71)(73)(79)(79)$

$p_{92} \ge 25$ with $420\times (17)(17)(43)(43)(49)(53)(61)(67)(67)(67)(67)(81)$

$p_{93} \ge 25$ with $840\times (19)(41)(41)(43)(47)(61)(67)(67)(67)(67)(81)(81)(89)$

$p_{94} \ge 26$ with $840\times (13)(19)(29)(29)(31)(59)(67)(71)(73)(73)(79)(89)(89)$

$p_{95} \ge 24$ with $420\times (25)(67)(67)(71)(71)(71)(73)(73)(73)(83)(89)(89)$

$p_{96} \ge 17$ with $15\times (23)(25)(25)(31)(31)(31)(31)(37)(41)(41)(43)(49)$

$p_{97} \ge 27$ with $112\times (25)(29)(31)(41)(43)(47)(49)(59)(73)(79)(81)(89)$

$p_{98} \ge 23$ with $4\times (17)(36)(47)(53)(53)(59)(59)(59)(71)(79)(79)(97)$

$p_{99} \ge 24$ with $96\times (37)(37)(49)(61)(64)(64)(64)(67)(89)(97)(97)(97)$

$p_{100} \ge 23$ with $7\times (11)(59)(59)(59)(59)(61)(61)(67)(67)(83)(83)(89)(89)(97)$