(In reply to
part one by Charlie)
The method used to find MH numbers in my first post will not prove part 2, that the list of such numbers is finite, for two reasons:
1. Testing every number in turn by increments of 1 would take "forever" for such large numbers as would be encountered.
2. We wouldn't know when to stop; there might always be more.
The solution would be to devise a recursive procedure that would start out at 1 and work its way up, with multiples taking the place of divisors, that is, actually being the divisors that would be used on the way down.
Every recursive procedure needs a stopping test so that it does not recurse forever. Of course if there were infinitely many MH numbers, the stopping procedure at some point would never be met, but at least there is a method in place so that eventually it does stop if the cardinality of the set is finite.
The below program's design is to do just that. For example it finds 400362681871113269704732351253634497228532915507726913665761280000, which is the 1017th number on the A114440 list and is a level 34 MH number down to 1. For each MH number that it finds, it tries all multiples of that number, starting at 2 times, until the multiplier (which acts as the divisor going in the other direction) is too large to be produced by summing the digits of the product (the potential MH number) even if all its digits were 9.
This particular implementation crashed when it was required to attempt a 50th level of recursion: apparently beyond the reach of UBASIC, at least for the currently written program. But, having faith in the OEIS statement of finitude, I'm sure the implementation in another language having the sufficient precision of arithmetic and also allowing a sufficient level of recursion, would terminate, thereby showing the list is finite, while, along the way, listing them, though not in order.
An aside: the termination via crash ocurred while processing numbers that eventually go through 7. Those that go through 2 through 6 are shown to be finite in number with, for example, only 54 and 972 going through 6.
4 kill "mhnumbrs.txt"
5 open "mhnumbrs.txt" for output as #2
10 gosub *MHN(1)
20
30 close #2
40 end
50
60 *MHN(n)
70 local mult,nextMH
80 mult=2:lvl=lvl+1
90 nextMH = n*mult
100 repeat
110 if fnSOD(nextMH) = mult then
120 :print lvl,nextMH
130 :print #2, lvl,nextMH
140 :gosub *MHN(nextMH)
150 inc mult
160 nextMH=n*mult
170 nMH=len(str(nextMH))
180 until nMH*9<mult
185 lvl=lvl-1
190 return
200
1070 fnSOD(X)
1080 Sod=0
1090 S=cutspc(str(X))
1100 for I=1 to len(S)
1110 Sod=Sod+val(mid(S,I,1))
1120 next
1130 return(Sod)
level MH number
1 2
2 18
3 162
4 2916
5 78732
6 2125764
7 76527504
8 3443737680
9 185961834720
10 10041939074880
10 11715595587360
11 632642161717440
12 45550235643655680
13 3689569087136110080
14 398473461410699888640
15 46621394985051886970880
16 5874295768116537758330880
17 845898590608781437199646720
18 98970135101227428152358666240
19 13360968238665702800568419942400
20 1923979426367861203281852471705600
21 259737222559661262443050083680256000
18 121809397047664526956749127680
12 56937794554569600
11 843522882289920
12 60733647524874240
13 4919425449514813440
14 531297948547599851520
15 52598496906212385300480
7 95659380
1 3
2 27
3 243
4 4374
5 118098
3 486
1 4
2 12
3 108
4 1944
5 52488
6 1417176
5 69984
2 24
3 216
2 36
3 324
4 5832
5 157464
6 4251528
6 5668704
7 153055008
4 8748
5 236196
3 648
4 11664
5 314928
6 8503056
4 17496
5 472392
5 629856
6 22674816
2 48
3 432
3 864
4 15552
5 419904
6 11337408
6 18895680
7 340122240
8 6122200320
9 220399211520
9 275499014400
9 330598817280
10 20827725488640
11 1499596235182080
12 121467295049748480
11 1687045764579840
12 91100471287311360
13 9838850899029626880
7 510183360
7 680244480
5 559872
6 15116544
1 5
2 45
3 405
4 7290
5 196830
6 7085880
7 255091680
8 9183300480
1 6
2 54
3 972
1 7
2 21
3 378
4 3402
5 61236
6 1102248
6 1653372
7 44641044
8 2008846980
9 108477736920
10 6834097425960
11 430548137835480
12 27124532683635240
13 2197087147374454440
7 59521392
8 2678462640
9 144636982560
10 6508664215200
10 7810397058240
11 421761441144960
11 562348588193280
8 3749847696
9 168743146320
9 202491775584
10 9112129901280
11 492055014669120
12 22142475660110400
13 1793540528468942400
12 35427961056176640
13 2550813196044718080
14 160701231350817239040
15 15909421903730906664960
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16 1874419162475932276162560
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11 574064183780640
12 36166043578180320
5 91854
6 2480058
4 6804
5 122472
6 3306744
7 119042784
8 4285540224
9 192849310080
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Posted by Charlie
on 2016-07-10 14:27:26 |
The method for part 2
