1.) 0
2.) 1, 1, 1, 1, 1, 1, 1, 1...
3.) 2, 2, 2, 2, 2, 2, 2, 2...
4.) 5, 5, 5, 5, 5, 5, 5, 5...
5.) 4, 8, 64, 2097152 ...
6.) 6, 36, 10077696 ...
7.) 7, 7, 7, 7, 7, 7, 7, 7...
8.) 9, 27, 729, 14348907 ...
9.) 10, 100, ____, ....
10.) 11, ____ ....

Another way of looking at this is that each successive number is the product of all the divisors of the preceding number, including that number itself (so you don't multiply that number by the product of the factors--it's just the product of the factor (divisors)--including the number itself).

A UBASIC program produces (when a number extends to the edge, it's continued on the next line--that's not a new number):

 2  2  2  2  2  2  2

 3  3  3  3  3  3  3

 4  8  64  2097152  345087317339528189371737793113851272622555448608519327758126
2111899648

 5  5  5  5  5  5  5

 6  36  10077696  14725249858351738531758616406923803289611205835143875798301476
96933725652329938814699489912177091370194610371452945920201737013751689122823344
84871403174767655880386045910084758703074415025299415269171925628511937032731984
01289798659170788350250654824767131118332645255421785880238670351400361139922403
6192306226210075368787167671873479497761345765376

 7  7  7  7  7  7  7

 8  64  2097152  345087317339528189371737793113851272622555448608519327758126211
1899648

 9  27  729  10460353203  164062694609488086547539746812648575659318856476507740
436897453222447961474036515706307096943652603236842951947

 10  100  1000000000  1000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000

 11  11  11  11  11  11  11

 12  1728  2116471057875484488839167999221661362284396544

 13  13  13  13  13  13  13

 14  196  20661046784  572289865518550906473459269984096192380710993219229673657
94652184423945333254122496651562041158958347909896818954965390353572266664573848
30774066987085123907460149528854520521907692049226249803601205254780949521935518
31514437559329125870814555210239068034820923267231914992058547700416695831022231
25005723472823193331239897610260659540671992240606962708382629504331967680571918
72156863597188577294362090789235477399989947165444379091109225494176959421892576
78736286644499109840378341193998665600647757288557433061376

where 9  27  729  10460353203  is the only discrepancy from the original puzzle; again, perhaps an error on the part of the poser.

It then confirms that 100 if followed by 1000000000 and 11 is followed by 11.

The program is:

    5   goto 1000
   10   *FactorIt
   20         N=abs(Num):if N>0 then Limit=sqrt(N):else Limit=0
   22         PSubscr=0
   30         if Limit<>int(Limit) then Limit=int(Limit+1)
   40         Dv=2:gosub *DivideIt
   50         Dv=3:gosub *DivideIt
   60         Dv=5:gosub *DivideIt
   70         Dv=7
   80         while 1
   90           gosub *DivideIt:Dv=Dv+4 '11
  100          gosub *DivideIt:Dv=Dv+2 '13
  110          gosub *DivideIt:Dv=Dv+4 '17
  120          gosub *DivideIt:Dv=Dv+2 '19
  130          gosub *DivideIt:Dv=Dv+4 '23
  140          gosub *DivideIt:Dv=Dv+6 '29
  150          gosub *DivideIt:Dv=Dv+2 '31
  160          gosub *DivideIt:Dv=Dv+6 '37
  170          if inkey=chr(27) then S$=chr(27):goto *Finis
  171          if Dv>Limit then goto *Outloop
  180         wend
  188       *Outloop
  190        if N>1 then
  191           :if N=PDivsr(PSubscr) then NOccur(PSubscr)=NOccur(PSubscr)+1:els
e PSubscr=PSubscr+1:PDivsr(PSubscr)=N:NOccur(PSubscr)=1:endif
  200    return

  220       *DivideIt
  230       while 1
  240         Q=int(N/Dv)
  250         if Q*Dv=N and N>0 then
  260           :N=Q:if N>0 then Limit=sqrt(N):else Limit=0:endif
  262           :if Dv=PDivsr(PSubscr) then NOccur(PSubscr)=NOccur(PSubscr)+1:el
se PSubscr=PSubscr+1:PDivsr(PSubscr)=Dv:NOccur(PSubscr)=1:endif
  270           :if Limit<>int(Limit) then Limit=int(Limit+1):else nop:endif
  280         :else
  290          :return
  300        :endif
  310       wend
  320        return

 1000   dim PDivsr(30)
 1010   dim NOccur(30):dim NTms(30)
 1020   for NStart=2 to 14
 1030     Num=NStart:print Num;
 1035     Iter=1
 1036     while Iter<=6
 1040      erase PDivsr():erase NOccur()
 1050      dim PDivsr(30):dim NOccur(30)
 1060      gosub *FactorIt
 1070      T=1:if PSubscr>0 then gosub *Analyze(1)
 1075      print T;:if T>10^20 then Iter=9
 1080      Num=T:Iter=Iter+1
 1190     wend
 1195   print:print
 1200   next NStart
 1300   end

 2000   *Analyze(S)
 2010   local NumTimes,I,J,Noc
 2090   Noc=NOccur(S)
 2100   for NumTimes=0 to Noc
 2110      NTms(S)=NumTimes
 2120      if S=PSubscr then
 2140        :for I=1 to PSubscr
 2150        :for J=1 to NTms(I)
 2160        :T=T*PDivsr(I)
 2170        :next J
 2180        :next I
 2190      :else
 2200        :gosub *Analyze(S+1)
 2215   next NumTimes
 2390   return

Edited on April 26, 2004, 3:42 pm

Posted by Charlie on 2004-04-26 15:40:42