# Loneliness of the Factorions

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### cliff

Apr 29, 1994, 3:42:27 PM4/29/94
to
Title: Cliff Puzzle 22: Factorions
From: cl...@watson.ibm.com

If you respond to this puzzle, if possible please include your name,
state, and e-mail address. If you like, tell me a little bit
about yourself. You might also directly mail me a copy of your response
in addition to any responding you do in the newsgroup. I will assume it
is OK to describe your answer in any article or publication I may write
in the future, with attribution to you, unless you state otherwise.
Thanks, Cliff Pickover
* * *

Factorions are numbers that are the sum of the factorial values for
each of their digits. For example, 145 is a factorion because
is can be expressed as 145 = 1! + 4! + 5!
The largest known factorion is 40,585 = 4! + 0! + 5! + 8! + 5!

Some mathematicians believe that this is the largest possible factorion.
(In fact, 145 and 40,585 are the only multi-digit factorions known to humanity.)

Can you end the loneliness of the factorions?
I would be interested in hearing from any of you who can find a larger
factorion for an article I am writing.

### John Scholes

May 1, 1994, 9:55:21 AM5/1/94
to
In article <2prnv3\$14...@watnews1.watson.ibm.com>
cl...@watson.ibm.com "cliff" writes:

>
> Factorions are numbers that are the sum of the factorial values for
> each of their digits. For example, 145 is a factorion because
> is can be expressed as 145 = 1! + 4! + 5!
> The largest known factorion is 40,585 = 4! + 0! + 5! + 8! + 5!
>
> Some mathematicians believe that this is the largest possible factorion.
> (In fact, 145 and 40,585 are the only multi-digit factorions known to
> humanity.)
>
> Can you end the loneliness of the factorions?
> I would be interested in hearing from any of you who can find a larger
> factorion for an article I am writing.
>
>

Unless I am missing something, this puzzle is pretty silly.
A factorion must have less than 8 digits, because 8 * 9! < 10^7.
A simple C program to check all such digits (time to write <5 minutes, time
to run < 1 minute) gives just 1,2,145 and 40585. No doubt one can debate
whether 1 and 2 qualify.

--
John Scholes

### Kevin S. Brown

May 1, 1994, 5:23:39 PM5/1/94
to
CW> Factorions are numbers that are the sum of the factorial values for
CW> each of their digits. For example, 145 is a factorion because is
CW> can be expressed as 145 = 1! + 4! + 5! The largest known factorion
CW> is 40,585 = 4! + 0! + 5! + 8! + 5! Some mathematicians believe that
CW> this is the largest possible factorion. (In fact, 145 and 40,585 are
CW> the only multi-digit factorions known to humanity.) Can you end the
CW> loneliness of the factorions? I would be interested in hearing from
CW> any of you who can find a larger factorion for an article I'm writing.

JS> Unless I am missing something, this puzzle is pretty silly.
JS> A factorion must have less than 8 digits, because 8 * 9! < 10^7.
JS> A simple C program...gives just 1,2,145 and 40585.

I don't think you are missing anything. It is known that these are the
only four integers equal to the sum of the factorials of their base 10
digits. This is discussed in Joe Roberts' book "The Lure of the Integers"
on page 35. He mentions a general theorem due to B. L. Schwartz stating
that the sum of a function f(n,d) of the base b digits of N is equal to N
for only a finite number of cases if

n [max f(n,d)]
lim sup ---------------- < 1 where d = 0 to b-1
n b^n

In addition to the simple example f(n,d) = d! for all n, he gives the more
interesting example f(n,d) = d^n. In this case the theorem tells us there
are only finitely many cases where an n-digit number is equal to the sum of
the nth powers of its digits. The known occurrences of this include 153,
1634, 8208, 9474, .... up to 4679307774. The theorem says there are only
finitely many of these, but it isn't known if 4679307774 is the largest.
(It's known that an upper bound for such a number is 10^60.)

### John Scholes

May 2, 1994, 7:04:11 PM5/2/94
to
In article <9405011722592....@delphi.com>
kevi...@delphi.com "Kevin S. Brown" writes:

> ... only finitely many cases where an n-digit number is equal to the sum of

> the nth powers of its digits. The known occurrences of this include 153,

> 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774
> is the largest ...

It doesn't seem too hard to find some more ...

32,164,049,650; 32,164,049,651; 40,028,394,225; 42,678,290,603;
44,708,635,679; 49,388,550,606; 82,693,916,578; 94,204,591,914

Any improvements, anyone?

--
John Scholes

### John Scholes

May 3, 1994, 5:28:42 AM5/3/94
to

In article <9405011722592....@delphi.com>
kevi...@delphi.com "Kevin S. Brown" writes:

> ... only finitely many cases where an n-digit number is equal to the sum of
> the nth powers of its digits. The known occurrences of this include 153,
> 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774
> is the largest ...

Another one: 28,116,440,335,967 (none with 12 or 13 digits).

--
John Scholes

### John Scholes

May 3, 1994, 12:25:26 PM5/3/94
to
In article <9405011722592....@delphi.com>
kevi...@delphi.com "Kevin S. Brown" writes:

> ... only finitely many cases where an n-digit number is equal to the sum of
> the nth powers of its digits. The known occurrences of this include 153,
> 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774
> is the largest ...

A few more: 16-digit 4,338,281,769,391,370 (& +1)
17-digit 21,897,142,587,612,075
35,641,594,208,964,132
35,875,699,062,250,035

These seem to take about half-an hour each on an elderly PC running
interpreted basic. So it looks as though it should be feasible with
a little more trouble to go all the way and find the largest.

--
John Scholes

### Dan Hoey

May 3, 1994, 4:38:13 PM5/3/94
to
kevi...@delphi.com "Kevin S. Brown" writes that there are

> ... only finitely many cases where an n-digit number is equal to the sum
> of the nth powers of its digits. The known occurrences of this include
> 153, 1634, 8208, 9474, .... up to 4679307774... but it isn't known if
> 4679307774 is the largest ...

According to David Wells's book _The Penguin Dictionary of Curious and
Interesting Numbers_ these numbers are called digital invariants. They
were mentioned on rec.puzzles in 1992, and Dik Winter provided a list from
1985 that I confirmed with a program I wrote. The following is a list of
all 89 base-ten digital invariants.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748,
92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050,
24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774,
32164049650, 32164049651, 40028394225, 42678290603, 44708635679,
49388550606, 82693916578, 94204591914, 28116440335967, 4338281769391370,
4338281769391371, 21897142587612075, 35641594208964132, 35875699062250035,
1517841543307505039, 3289582984443187032, 4498128791164624869,
4929273885928088826, 63105425988599693916, 128468643043731391252,
449177399146038697307, 21887696841122916288858, 27879694893054074471405,
27907865009977052567814, 28361281321319229463398, 35452590104031691935943,
174088005938065293023722, 188451485447897896036875,
239313664430041569350093, 1550475334214501539088894,
1553242162893771850669378, 3706907995955475988644380,
3706907995955475988644381, 4422095118095899619457938,
121204998563613372405438066, 121270696006801314328439376,
128851796696487777842012787, 174650464499531377631639254,
177265453171792792366489765, 14607640612971980372614873089,
19008174136254279995012734740, 19008174136254279995012734741,
23866716435523975980390369295, 1145037275765491025924292050346,
1927890457142960697580636236639, 2309092682616190307509695338915,
17333509997782249308725103962772, 186709961001538790100634132976990,
186709961001538790100634132976991, 1122763285329372541592822900204593,
12639369517103790328947807201478392, 12679937780272278566303885594196922,
1219167219625434121569735803609966019,
12815792078366059955099770545296129367,
115132219018763992565095597973971522400, and
115132219018763992565095597973971522401.

My program checked that there are no more up to 60 digits, and it is easy
to prove that there can be none larger.

As a devout zerophilist, I of course include zero on the list. While I
believe that zero is, properly speaking, not a one-digit number but a
zero-digit number (customarily written with a leading zero), it is a
digital invariant in either case.

Dan Hoey
Ho...@AIC.NRL.Navy.Mil

### Dik T. Winter

May 3, 1994, 7:30:45 PM5/3/94
to
In article <767982...@kalva.demon.co.uk> jsch...@kalva.demon.co.uk writes:
> In article <9405011722592....@delphi.com>
> kevi...@delphi.com "Kevin S. Brown" writes:
>
> > ... only finitely many cases where an n-digit number is equal to the sum of
> > the nth powers of its digits. The known occurrences of this include 153,
> > 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774
> > is the largest ...
>
> A few more: 16-digit 4,338,281,769,391,370 (& +1)
> 17-digit 21,897,142,587,612,075
> 35,641,594,208,964,132
> 35,875,699,062,250,035
>
1 digit (of course)
1; 2; 3; 4; 5; 6; 7; 8; 9
3 digits
153; 370; 371; 407
4 digits
1634; 8208; 9474
5 digits
54748; 92727; 93084
6 digits
548834
7 digits
1741725; 4210818; 9800817; 9926315
8 digits
24678050; 24678051; 88593477
9 digits
146511208; 472335975; 534494836; 912985153
10 digits
4679307774
11 digits
32164049650; 32164049651; 40028394225; 42678290603; 44708635679; 49388550606
82693916578; 94204591914
14 digits
28116440335967
16 digits
4338281769391370; 4338281769391371
17 digits
21897142587612075; 35641594208964132; 35875699062250035
19 digits
1517841543307505039; 3289582984443187032; 4498128791164624869
4929273885928088826
20 digits
63105425988599693916
21 digits
128468643043731391252; 449177399146038697307
23 digits
21887696841122916288858; 27879694893054074471405; 27907865009977052567814
28361281321319229463398; 35452590104031691935943
24 digits
174088005938065293023722; 188451485447897896036875; 239313664430041569350093
25 digits
1550475334214501539088894; 1553242162893771850669378; 3706907995955475988644380
3706907995955475988644381; 4422095118095899619457938
27 digits
121204998563613372405438066; 121270696006801314328439376
128851796696487777842012787; 174650464499531377631639254
177265453171792792366489765
29 digits
14607640612971980372614873089; 19008174136254279995012734740
19008174136254279995012734741; 23866716435523975980390369295
31 digits
1145037275765491025924292050346; 1927890457142960697580636236639
2309092682616190307509695338915
32 digits
17333509997782249308725103962772
33 digits
186709961001538790100634132976990; 186709961001538790100634132976991
34 digits
1122763285329372541592822900204593
35 digits
12639369517103790328947807201478392; 12679937780272278566303885594196922
37 digits
1219167219625434121569735803609966019
38 digits
12815792078366059955099770545296129367
39 digits
115132219018763992565095597973971522400
115132219018763992565095597973971522401

If I am right this took about 2000 seconds on a Vax back in 1985.
Doing the same base 12 took 60 times as much. (I think those were
seconds indeed, otherwise I would never have done base 12.)

Here follows the total number of solutions (including one digit
solutions) for the different bases:

base total# largest (preceded by # digits in parenthesis)
2 1 ( 1) 1
3 5 ( 3) 122
4 11 ( 4) 3303
5 17 (14) 14421440424444
6 30 (18) 105144341423554535
7 59 (23) 12616604301406016036306
8 62 (29) 11254613377540170731271074472
9 58 (30) 104836124432728001478001038311
10 88 (39) 115132219018763992565095597973971522401
11 134 (45) 12344AA12A721803422912A8AA4963568083A268456A4
12 87 (51) 15079346A6B3B14BB56B395898B96629A8B01515344B4B0714B

Now that computers are so much faster I might try to expand the list.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098
home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: d...@cwi.nl

### Erich Friedman

May 4, 1994, 10:46:33 AM5/4/94
to
In article Dan Hoey writes:

> According to David Wells's book _The Penguin Dictionary of Curious and
> Interesting Numbers_ these numbers are called digital invariants.

Is this book still in print?

Erich Friedman
frie...@macs.stetson.edu

### david...@delphi.com

May 7, 1994, 9:36:00 PM5/7/94
to
Yes it is. I got a copy about a year ago. About ten dollars. The
US address is Penguin Books, 375 Hudson Street, NY,NY, 10014.
Or you can call the 800 operator and ask for the 800 number for
Penguin Books.

### Tomas Antonio Mendes Oliveira e Silva

May 9, 1994, 11:32:54 AM5/9/94
to
Kevin S. Brown (kevi...@delphi.com) wrote:
: CW> Factorions are numbers that are the sum of the factorial values for

Here goes the complete list of integers for this last case [ f(n,d) = d^n ].
Incidently, they are usually called Plus Perfect Numbers, and can be computed
for bases other that 10 (for any given base there is always a finite number of
them).

0

1
2
3
4
5
6
7
8
9

153
370
371
407
1634
8208
9474
54748
92727
93084
548834
1741725
4210818
9800817
9926315
24678050
24678051
88593477
146511208
472335975
534494836
912985153
4679307774

32164049650
32164049651
40028394225
42678290603
44708635679
49388550606
82693916578
94204591914

28116440335967
4338281769391370
4338281769391371
21897142587612075
35641594208964132
35875699062250035
1517841543307505039
3289582984443187032
4498128791164624869
4929273885928088826
63105425988599693916
128468643043731391252
449177399146038697307

21887696841122916288858
27879694893054074471405
27907865009977052567814
28361281321319229463398
35452590104031691935943

174088005938065293023722
188451485447897896036875
239313664430041569350093

1550475334214501539088894
1553242162893771850669378
3706907995955475988644380
3706907995955475988644381
4422095118095899619457938

121204998563613372405438066
121270696006801314328439376
128851796696487777842012787
174650464499531377631639254
177265453171792792366489765

14607640612971980372614873089
19008174136254279995012734740
19008174136254279995012734741
23866716435523975980390369295
1145037275765491025924292050346
1927890457142960697580636236639
2309092682616190307509695338915
17333509997782249308725103962772
186709961001538790100634132976990
186709961001538790100634132976991
1122763285329372541592822900204593
12639369517103790328947807201478392
12679937780272278566303885594196922
1219167219625434121569735803609966019
12815792078366059955099770545296129367
115132219018763992565095597973971522400
115132219018763992565095597973971522401

I can send the C source code of the program that generated this list to anyone
interested. Since it was one of the first programs that I wrote in this
language don't expect too much in terms of readability.

*****************************************************
* Tomas Oliveira e Silva *
* INESC Aveiro/DETUA *