mtj@ncvet.UUCP (Michael Jones) (10/31/84)
The first Data Systems Research, Inc. contest has been won by Fred Helenius
of Boston, MA. His winning entry was postmarked October 17, 1984. Quite a
few entrants failed to find the proper solution, and none appeared aware of
the name for this sequence.
The information given in the contest announcement was:
1. The following four numbers are part of a much longer sequence.
54748
92727
93084
548834
2. They appear in this longer sequence consecutively.
3. They appear in this longer sequence in the order given (top to bottom).
4. Each number appears only once in the longer sequence.
5. The longer sequence consists of numbers with a special property.
6. The longer sequence is finite.
The questions asked were:
1. What two numbers (if any) preceed 54748 in the longer sequence ?
2. What two numbers (if any) succeed 548834 in the longer sequence ?
3. What special property do these numbers possess ?
The answers are:
1. 8208 and 9474 preceed 54748 in the longer sequence.
2. 1741725 and 4210818 succeed 548834 in the longer sequence.
3. These numbers are n-digit numbers which are equal to the sum
of the n-th powers of their digits.
Some numbers in this sequence are:
153 = 1^3 + 5^3 + 3^3
8208 = 8^4 + 2^4 + 0^4 + 8^4
54748 = 5^5 + 4^5 + 7^5 + 4^5 + 8^5
548834 = 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6
4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7
A very large number with this property is:
128468643043731391252 = 1^21 + 2^21 + 8^21 + ... + 5^21 + 2^21
This first Data Systems Research, Inc. contest has been fun for us, and
for the many of you who entered. The second contest will begin soon. It
will be announced on the net, as well as on personal RBBS systems.dik@turing.UUCP (11/17/84)
Well some time ago the referenced article appeared, and many among us will
have pondered about the generation of the complete set of solutions to
the problem. Here it follows. The solutions were generated in slightly less
than 2000 seconds on a CDC Cyber 170-750 (about 15.5 times as fast as a
DEC Vax 11/750).
(In summary the problem is to find n digit numbers were the number is the
sum of the n-th powers of its digits.)
First a table displaying for a number of digits the number of solutions.
There are no solutions of 40 or more digits.
# digs # sols # digs #sols # digs # sols
1 9 14 1 27 5
2 - 15 - 28 -
3 4 16 2 29 4
4 3 17 3 30 -
5 3 18 - 31 3
6 1 19 4 32 1
7 4 20 1 33 2
8 3 21 2 34 1
9 4 22 - 35 2
10 1 23 5 36 -
11 8 24 3 37 1
12 - 25 5 38 1
13 - 26 - 39 2
So there are 88 solutions. It may be seen that the majority of the
solutions has an odd number of digits (70 odd, 18 even) even if we
ignore the (trivial) one-digit solutions. This appears not to be
accidental, because the same holds if we pose the problem for other
bases than base 10. Is there someone who has an explanation for this?
The largest solution is the 39-digit number:
115132219018763992565095597973971522401
The complete list of solutions:
n = 1
1
2
3
4
5
6
7
8
9
n = 3
153
370
371
407
n = 4
1634
8208
9474
n = 5
54748
92727
93084
n = 6
548834
n = 7
1741725
4210818
9800817
9926315
n = 8
24678050
24678051
88593477
n = 9
146511208
472335975
534494836
912985153
n = 10
4679307774
n = 11
32164049650
32164049651
40028394225
42678290603
44708635679
49388550606
82693916578
94204591914
n = 14
28116440335967
n = 16
4338281769391370
4338281769391371
n = 17
21897142587612075
35641594208964132
35875699062250035
n = 19
1517841543307505039
3289582984443187032
4498128791164624869
4929273885928088826
n = 20
63105425988599693916
n = 21
128468643043731391252
449177399146038697307
n = 23
21887696841122916288858
27879694893054074471405
27907865009977052567814
28361281321319229463398
35452590104031691935943
n = 24
174088005938065293023722
188451485447897896036875
239313664430041569350093
n = 25
1550475334214501539088894
1553242162893771850669378
3706907995955475988644380
3706907995955475988644381
4422095118095899619457938
n = 27
121204998563613372405438066
121270696006801314328439376
128851796696487777842012787
174650464499531377631639254
177265453171792792366489765
n = 29
14607640612971980372614873089
19008174136254279995012734740
19008174136254279995012734741
23866716435523975980390369295
n = 31
1145037275765491025924292050346
1927890457142960697580636236639
2309092682616190307509695338915
n = 32
17333509997782249308725103962772
n = 33
186709961001538790100634132976990
186709961001538790100634132976991
n = 34
1122763285329372541592822900204593
n = 35
12639369517103790328947807201478392
12679937780272278566303885594196922
n = 37
1219167219625434121569735803609966019
n = 38
12815792078366059955099770545296129367
n = 39
115132219018763992565095597973971522400
115132219018763992565095597973971522401
--
dik t. winter
centrum voor wiskunde en informatica
postbus 4079
1009 AB amsterdam
nederland
+31 20 592 4102 (polish your dutch)
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