Let f(n) denote the largest prime factor of n which is not a palindrome.
Find all the numbers n (below 20000) such that the sum of the factorials of the digits of n is equal to f(n) reversed.
Example: 2105 = 5*421 and 2!+1!+0!+5! = 124.
10 for N=2 to 20000
20 Ffor=cutspc(str(fnF(N)))
30 Frev=""
40 for I=1 to len(Ffor)
50 Frev=mid(Ffor,I,1)+Frev
60 next I
70 Nfor=cutspc(str(N)):Tot=0
80 for I=1 to len(Nfor)
90 Tot=Tot+!(val(mid(Nfor,I,1)))
100 next I
110 if Tot=val(Frev) then print N,Ffor,Frev,Tot
120 :gosub *Listpf:gosub *Listfacts:print:inc Solct
130 next
140 print Solct
150 end
160
170
180 fnF(N)
190 local Tr,Lgfact
200 Tr=N:Lgfact=1
210 while Tr>1
220 Pf=prmdiv(Tr)
230 if fnPalin(Pf)=0 then
240 :if Pf>Lgfact then Lgfact=Pf
250 Tr=Tr//Pf
260 wend
270 return(Lgfact)
280
290 fnPalin(X)
300 local S,I,Good
310 S=cutspc(str(X))
320 Good=1
330 for I=1 to int(len(S)/2)
340 if mid(S,I,1)<>mid(S,len(S)+1-I,1) then Good=0
350 next
360 return(Good)
400 '
410 *Listpf
420 Tr=N:print " ";
430 while Tr>1
440 Pf=prmdiv(Tr):print Pf;:Tr=Tr//Pf
450 wend:print
460 return
500 '
510 *Listfacts
515 print " ";
520 for I=1 to len(Nfor):print !(val(mid(Nfor,I,1)));:next
530 print:return
In each of the 12 sets below, the first row has the number, n, f(n), f(n) reversed and the total of the factorials of the digits.
For verification, the second line lists all the prime factors and the third line lists the factorial of each digit in turn.
143 13 31 31
11 13
1 24 6
541 541 145 145
541
120 24 1
2105 421 124 124
5 421
2 1 1 120
2444 47 74 74
2 2 13 47
2 24 24 24
3431 73 37 37
47 73
6 24 6 1
4144 37 73 73
2 2 2 2 7 37
24 1 24 24
4233 83 38 38
3 17 83
24 2 6 6
4301 23 32 32
11 17 23
24 6 1 1
4440 37 73 73
2 2 2 3 5 37
24 24 24 1
10234 43 34 34
2 7 17 43
1 1 2 6 24
12243 53 35 35
3 7 11 53
1 2 2 24 6
12341 43 34 34
7 41 43
1 2 6 24 1
Since these seemed to be concentrated in the beginning of the first 20,000, I thought it might be good to see if there were none beyond; but there are, starting with:
20313 61 16 16
3 3 37 61
2 1 6 1 6
22320 31 13 13
2 2 2 2 3 3 5 31
2 2 6 2 1
30422 53 35 35
2 7 41 53
6 1 24 2 2
34030 83 38 38
2 5 41 83
6 24 1 6 1
34144 97 79 79
2 2 2 2 2 11 97
6 24 1 24 24
35140 251 152 152
2 2 5 7 251
6 120 1 24 1
46003 257 752 752
179 257
24 720 1 1 6
52100 521 125 125
2 2 5 5 521
120 2 1 1 1
53013 431 134 134
3 41 431
120 6 1 1 6
102613 137 731 731
7 107 137
1 1 2 720 1 6
106312 137 731 731
2 2 2 97 137
1 1 720 6 1 2
112413 53 35 35
3 7 53 101
1 1 2 24 1 6
113162 137 731 731
2 7 59 137
1 1 6 1 720 2
120032 31 13 13
2 2 2 2 2 11 11 31
1 2 1 1 6 2
134046 677 776 776
2 3 3 11 677
1 6 24 1 24 720
200340 53 35 35
2 2 3 3 3 5 7 53
2 1 1 6 24 1
202124 23 32 32
2 2 13 13 13 23
2 1 2 1 2 24
203112 31 13 13
2 2 2 3 3 7 13 31
2 1 6 1 1 2
210304 53 35 35
2 2 2 2 2 2 2 31 53
2 1 1 6 1 24
211203 31 13 13
3 3 31 757
2 1 1 2 1 6
211232 41 14 14
2 2 2 2 2 7 23 41
2 1 1 2 6 2
212004 13 31 31
2 2 3 3 3 13 151
2 1 2 1 1 24
212264 157 751 751
2 2 2 13 13 157
2 1 2 2 720 24
221030 31 13 13
2 5 23 31 31
2 2 1 1 6 1
222224 43 34 34
2 2 2 2 17 19 43
2 2 2 2 2 24
230160 137 731 731
2 2 2 2 3 5 7 137
2 6 1 1 720 1
230315 631 136 136
5 73 631
2 6 1 6 1 120
231012 31 13 13
2 2 3 3 3 3 23 31
2 6 1 1 1 2
234354 281 182 182
2 3 139 281
2 6 24 6 120 24
240212 23 32 32
2 2 7 23 373
2 24 1 2 1 2
250404 271 172 172
2 2 3 7 11 271
2 120 1 24 1 24
253031 631 136 136
401 631
2 120 6 1 6 1
260300 137 731 731
2 2 5 5 19 137
2 720 1 6 1 1
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Posted by Charlie
on 2014-02-27 17:05:19 |
computer solution
