Joseph L. Pe

iDEN System Engineering Tools and Statistics

Motorola

1421 W. Shure Drive

Room S151

Arlington Heights, IL 60004

email: ajp070@motorola.com

**1.
****Introduction**

** **

Picture-perfect numbers (ppn’s), a
species of the generalized perfect numbers introduced in [P],* *are defined as the numbers n such that
the reverse of n equals the sum of the reverses of the proper divisors of n.
This article tells the story of a concerted search for ppn’s, leading to the
discovery of a structure theorem for these numbers. The theorem, called *Andersen’s
theorem* after its discoverer, states that the number *p* = 140{0}* _{z}*10{9}

In this discussion, *number *refers
to natural number. Functions are called *arithmetical *if their domains
are sets of integers. For simplicity, an arithmetical function in this article
will also be integer-valued and have the set of natural numbers as its domain.
Unless otherwise noted, sequence numbers (which begin A…) refer to those used
in [S].

**2.
****Perfect Number Recreations**

** **

The paper [P] introduced
generalized perfect numbers. If *f* is an arithmetical function, then the
set of *f*-perfect numbers is the collection of numbers *n* such that
*f*(*n*) equals the sum of *f*(*d*),
where *d* ranges over all proper divisors of *n*; symbolically,

While this generalization adds a
new perspective to the question of the existence of odd perfect numbers (odd
numbers appear late in perfect number sequences such as A066229, where *f*(*n*)
= *n* + 1), it can also provide amusing and curious relations for *f*
that transforms its argument’s digits in some way.

As a quick example, let *f*(*n*)
= *n*<>*n*, the concatenation of *n* with itself (e.g., *f*(123)
= 123123). Then 6 is an *f*-perfect number (indeed, it is the only such
number less than 10^{6}), and also 6 is the first perfect number. This
observation yields the pretty pair of equations

6 = 1 + 2 + 3

66 = 11 + 22 + 33.

Let *R*(*n*) denote the
reverse of *n*, with leading zeros ignored (e.g., *R*(123) = 321 and *R*(120)
= 21). In the next section, the *f*-perfect
numbers resulting from *f*(*n*) = *R*(*n*), will be
examined.

** **

**3. The Elusive
Picture-Perfect Numbers**

** **

There is something very special about the number 10311.

Recall that a number is perfect if it equals the sum of its proper divisors. 10311 is not perfect: its proper divisors are 1, 3, 7, 21, 491, 1473, 3437, which sum to 5433. (Since 5433 < 10311, 10311 is said to be "deficient".) Hence, the equation

10311 = 1 + 3 + 7 + 21 + 491 + 1473 + 3437

is ** not**
valid. However, reading this invalid equation

7343 + 3741 + 194 + 12 + 7 + 3 + 1 = 11301

which, amazingly, is valid!

A number *n* is
called *picture-perfect* or *mirror-perfect *if the reverse of *n*
equals the sum of the reverses of the proper divisors of *n*. (Ignore
leading 0s.) In short, *n* is an *f*-perfect number, where *f*(*n*)
= *R*(*n*). If *n* is placed on one side of an equality sign and
the (unevaluated) sum of the proper divisors of *n* are placed on the other
side, then the resulting equation read backwards is valid. This explains the
use of the term “picture-perfect”, since a picture of an object is a mirror
image (i.e., an orientation reversal) of that object.

The first picture-perfect number (ppn, for short) is 6, the first perfect number; but since 6 and its proper divisors are single digit, it is trivially picture-perfect. The first non-trivial ppn is 10311.

If *P* is a
ppn, then call the expression *P* =_{b}*D*, where *D* is the (unevaluated) sum
of the proper divisors of *P*, the *mirror equation* of *P*.
The symbol =* _{b}* indicates that the mirror equation
should be read backwards to be valid. For example, the mirror equation of 10311
is

10311 =* _{b}* 1 + 3 + 7 + 21
+ 491 + 1473 + 3437.

Here is *Mathematica*
code to generate picture-perfect numbers not exceeding 10^{10}:

f[n_] :=
FromDigits[Reverse[IntegerDigits[n]]];

n
= 2; (*Initial value of n*)

While[n<10^10, `If[f[n]==Apply[Plus,Map[f,`

Drop[Divisors[n], -1]]], Print[n]]; n++]

When my computer search
had turned up no new ppn’s less than 10^{7}, I was about to conjecture
that 10311 was the only non-trivial such number. However, after several hours
of *Mathematica* running on my machine, I was rewarded with the third ppn:
21661371, which has mirror equation

21661371 =* _{b}* 1 + 3 + 9 +
27 + 443 + 1329 + 1811 + 3987 + 5433 + 11961 +

16299 + 48897 + 802273 + 2406819 + 7220457.

The discovery of this large ppn raised hopes that a fourth number would be found before long. Two problem enthusiasts, Daniel Dockery and Mark Ganson, soon joined me in the search. Our group has a discussion forum [Su] which we use to share ideas and results, as well as software customized for ppn search. For example, Ganson provides a Windows search utility he has written. The results and conjectures mentioned in this article first appeared in [Su].

We focused our efforts on the interval from 10^{9}
to 10^{10}. Three weeks later, Dockery found the fourth ppn. It is the
ten-digit 1460501511, with mirror equation

1460501511 =* _{b}* 1 + 3 + 7 + 21 + 101 + 303 + 707 + 2121
+ 688591 +

2065773 + 4820137 + 14460411 + 69547691 + 208643073 + 486833837.

After this find, our progress was slow, even though
Ganson’s C++ search program ran at least twice as fast as *Mathematica*.
Then Jens Kruse Andersen joined the team by announcing his discovery of a new
ppn: 7980062073 with mirror equation

7980062073 =_{b} 1 + 3 + 19 + 57 + 140001089 + 420003267
+ 2660020691.

Devising an efficient algorithm
that caches divisor information, Andersen quickly tested all numbers up to 10^{10},
and concluded that there are no more ppn’s in this range other than the five
already mentioned above. Within a month, our team exhaustively searched the
interval from 10^{10 }to 10^{12 }using Andersen’s program.

The sequence of ppn’s

6, 10311, 21661371, 1460501511, 7980062073, 79862699373, 798006269373, …

is sequence A069942 of [S], and
also appears in its short index. “Small” ppn’s are rare pearls in the infinite
ocean of numbers—there are only five ppn’s below 10^{10}. The seven
ppn’s listed above are all the ppn’s less than 10^{12}. At the time of
writing, the value of the eighth ppn is unknown. Unexpectedly, this sparseness
is somewhat relieved in the realm of large numbers, as the next section will
show.

**4. Andersen’s Theorem**

where *n* is a prime such that
(2* ^{n}*-1)

Andersen discovered the remarkable
result that **if the decimal number p =
140 z10n89 is prime, then the product 57 p is
picture-perfect, and conversely**, where

Call a prime *p* of the form
140*z*10*n*89 an *Andersen prime*, and its corresponding
picture-perfect number 57 *p* an *Andersen number*. Examples of
Andersen prime/ number pairs are:

__Andersen
Prime Corresponding
Andersen Number__

* *

140**00**1089 798**00**62073

14010**999**89 798626**99**373

140**00**10**99**89 798**00**626**9**373

140**000**10**99999**89 798**000**626**9999**373

The finite sequences *z*, *n*,
and *m* (as defined above) appear in **boldface**.

Soon after Andersen’s announcement
of his result, Ganson used Andersen’s theorem to identify 204 (mostly huge) Andersen
numbers, the largest with 177 digits. This is the gargantuan 798*z*626*m*373
where *z* has 77 zeros and
*m* has 91 nines.
Eventually, Ganson and Andersen discovered thousands of Andersen primes. At the
time of writing, the largest Andersen prime known is the 2461-digit 140 x 10^{2458
}+ 1089, verified as prime by Andersen using the program *Primo* by
Marcel Martin. Its corresponding Andersen number is the 2462-digit 798 x 10^{2459
}+ 62073. Andersen and Ganson have also found probable Andersen primes
with more than 10^{4} digits, but there currently seems to be no way to
prove primality for numbers of this size.

The sequence of picture-perfect numbers is an example of a sequence that appears at first to be extremely sparse—even finite—but yields many terms in the scale of the very large. Indeed, the sequence of ppn’s has been compared to what first appears to be a faint star in the universe of numbers, but is then revealed to be an abundant galaxy by the computer-telescope. An open problem is whether picture-perfect numbers such as 10311 can generate other picture-perfect numbers in the same way as Andersen’s 7980062073 does.

The proof of Andersen’s theorem now follows. The proof is due to Andersen, except for the proof of Andersen’s lemma, which I have provided.

**Andersen’s Lemma** If *p* (not necessarily prime) is of the
abovementioned form 140*z*10*n*89, then *R*(57 *p*) = 170 +
*R*(p) + *R*(3 *p*) + *R*(19 *p*).

Assuming for the meantime the validity of the
lemma, suppose that *p* is prime. Then the proper divisors of 57 *p*
= 3 x 19 x *p* are

1,
3, 19, 57, *p*, 3 *p*, 19 *p*. (D)

Adding the reverses of these proper divisors,

*R*(1) + *R*(3)
+ *R*(19) + *R*(57) + *R*(*p*) + *R*(3 *p*) + *R*(19
*p*)

= 170 + *R*(*p*)
+ *R*(3 *p*) + *R*(19 *p*)

= *R*(57 *p*),
by the lemma.

Hence, 57 *p*
is picture-perfect.

Conversely, if p is not prime, then 57 *p*
will have more divisors than (D) above, and the sum of the reversed divisors
becomes larger than 170 + *R*(*p*) + *R*(3 *p*) + *R*(19
*p*) = *R*(57 *p*). Hence, 57 *p* is not picture-perfect.
This completes the proof of Andersen’s theorem.

To establish Andersen’s lemma, consider two cases.

*Case (1)* *n* has at least one 9, that
is, *p* = 140*z*10*n*89. In what follows, the finite sequences *z*,
*n*, and *m* (as defined above) appear in **boldface**. To
simplify notation (which can easily get cluttered with subscripts), *z*, *n*,
*m* appear as **00**, **999**, and **99**, respectively. There is no loss of generality here: the boldfaced
finite sequences can be replaced by arbitrary sequences in the appropriate
manner without affecting the correctness of the proof. For example, everywhere
in the proof, **00 **(i.e., *z*) and **999 **(i.e., *n*) can be
replaced by, say, **0000** and **9999**, respectively, in which case, **99**
(i.e., *m*) must then be replaced by **999**.

One should convince oneself of the generality of the argument by performing the arithmetical operations below by hand. In doing so, one gains a better understanding of the numerical patterns at work here.

Thus, with *p*
= 140**00**10**999**89, simple multiplication shows that

57 *p* = 798**00**626**99**373 *R*(57 *p*) = 373**99**626**00**897

19 *p* = 266**00**208**99**791 *R*(19 *p*) = 197**99**802**00**662

3 *p* = 420**00**32**999**67 *R*(3 *p*) = 76**999**23**00**024

*p* = 140**00**10**999**89 *R*(*p*) = 98**999**01**00**041

Adding *R*(19 *p*), *R*(3
*p*), *R*(*p*), and 170 gives *R*(57 *p*), as

required:

197**99**802**00**662

+
76**999**23**00**024

98**999**01**00**041

__ 170__

373**99**626**00**897

*Case (2)* *n* has no 9s,
that is, *p* = 140*z*1089. The proof here is handled similarly as in
case (1). I only mention the final summation of *R*(19 *p*), *R*(3
*p*), *R*(*p*), and 170 to *R*(57 *p*), using the same
notation as case (1):

19602**00**662

+
7623**00**024

9801**00**041

__ 170__

37026**00**897

**5.
****Conjectures and Extensions**

** **

Recently, Andersen
discovered an extension of his theorem: *p* = 140{{0}_{z}_{(k)}10{9}_{n}_{(k)}89}* _{k}*
is prime if and only if 57

** **

** **Andersen
conjectures that there are infinitely many Andersen primes (for both his

theorem and its extension), hence infinitely many corresponding Andersen

numbers.

Even before the
third number 21661371 had been found, Ganson had conjectured that all
picture-perfect numbers are divisible by 3. Of course, the discoveries of
21661371, 1460501511, and the many Andersen numbers, all multiples of 3, have
further strengthened *Ganson’s conjecture*.

Since the ancient Greeks first posed it, the conjecture that every perfect number is even has remained unanswered. Of course, in the “mirror”, things appear reversed. Ganson and I conjecture that every non-trivial picture-perfect number is odd. (6, the only trivial picture-perfect number, is even.) We also believe that 6 is the only number that is both perfect and picture-perfect.

Ganson investigated
picture-perfect numbers in bases other than 10. In base 6, he found only four
such numbers not exceeding 10^{6}: 28 = 44_{6}, 145 = 401_{6},
901 = 4101_{6}, and 1081 = 5001_{6}. For example, 145 = 401_{6
}has mirror equation

401_{6} =_{b} 45_{6}
+ 5_{6 }+ 1_{6},

where the sum on the right-hand side of the equation is a base-6 sum.

More intriguingly,
Ganson found 41 base-5 picture-perfect numbers less than 10^{6}.

Continuing the computation, Dockery found 38 more such numbers less than

2.1 x 10^{7}.
All but one of these 79 base-5 numbers are divisible by 3. The single exception
is 5. However, Ganson notes that for any prime *p*, *p* has only one
proper divisor (i.e., 1) and has a base-*p* representation as 10;
therefore *p* is trivially base-*p* picture-perfect. 5 is trivially
base-5 picture-perfect. Thus, Ganson conjectures that every non-trivial* *base-5
picture-perfect number is divisible by 3.

Observe that a
perfect number *n* is (trivially) picture-perfect in any base *b*
larger than *n*. This is because n and its proper divisors have one-digit
base-*b* representations. For example, 6 and 28 are picture-perfect in
base 29. A perfect number *n* is also picture-perfect in base *n*-1,
where *n* is represented as the palindronomic 11 and all proper divisors
of *n* are single-digit, so reversing changes nothing.

Here is the *Mathematica
*code Ganson used to investigate the situation for other bases:

f[n_] := FromDigits[Reverse[IntegerDigits[n, base]], base];

baseDivisors[n_, base_] :=
IntegerDigits[Drop[Divisors[n], -1], base];

Do[startFrom = 2;Do[If[f[n] ==
Apply[Plus, Map[f, Drop[Divisors[n], -1]]],

Print["base = ", base,
", n = ", n, ") ", IntegerDigits[n, base],

" divisors: ",
Drop[Divisors[n], -1], " base divisors: ",

baseDivisors[n, base]]], {n,
startFrom, 10000}], {base, 2, 10}]

To conclude this section, I mention
two variations of the ppn sequence. A number *n* is called *tcefrep*
(“perfect” spelled backwards) if *R*(*n*) = the sum of the proper
divisors of *n*. *n* is called *anti-perfect* if *n* = the
sum of the reverses of the proper divisors of *n*.

The first five tcefrep numbers are

6, 498906, 20671542, 41673714, 73687923

(Sequence A072234.) Andersen discovered that 4158499614 is tcefrep,

although it might not be the sixth term. Ganson conjectured that all tcefrep

numbers are divisible by 3 (“Nosnag’s” conjecture).

On the other hand, no obvious pattern applies to the first five anti-perfect numbers

6,
244, 285, 133857, 141635817

(Sequence A072228.) Andersen, who found the fifth term, checked that

these are all the terms less than 10^{10}.

** **

**6. Picture-Perfect Semi-Primes**

Little is
known about the ppn’s in general, especially those not generated by Andersen’s
theorem. In investigating ppn’s, it is natural to start with numbers having few
prime factors. Obviously, no prime can be a ppn. Hence, the simplest possible
ppn’s are the semi-prime ppn’s, that is, ppn’s with exactly two prime factors.
6 = 2 x 3 is the smallest semi-prime ppn and the only one currently known.

Ganson asked if a number of the form 3 *p*, with *p* > 3
prime, can be a ppn. I answered this question in the negative; the proof now
follows.

If such a prime *p* did exist,
then the proper divisors of *n* = 3 *p* are: 1, 3, *p*. By
definition,

*R*(3 *p*) = *R*(1) + *R*(3) + *R*(*p*),

that is,

*R*(3 *p*) - *R*(*p*) = 4. (*)

By simple exhaustive computation, any such *p* > 2 must have at least
three digits. Only two cases are possible:

Case 1. *R*(*p*) and *R*(3 *p*) have the same number of digits.
By (*), the first digits (starting from the left) of *R*(*p*) and *R*(3
*p*) must differ by at most 1. Reversing, we see that the last digits of *p*
and 3 *p* must differ by at most 1. The only way this can be satisfied is
for the last digit of *p* to be 0 or 5. Obviously, this is a
contradiction, since a prime cannot end in 0 or 5.

Case 2. *R*(3 *p*) has one more digit than *R*(*p*). Then
by (*), the first digit of *R*(3 *p*) is 1 and the first digit of *R*(*p*)
is 9. Reversing, the last digit of 3 *p* is 1 and the last digit of *p*
is 9. This is a contradiction since the last digit of 3 *p* would have to
be 7 if the last digit of p was 9.

In any case, a contradiction is reached.

I also discovered the following
result on semi-prime ppn’s: There is no
ppn

semi-prime with each of its prime factors having at least four digits.
That is to say, any ppn semi-prime has a prime factor less than 1000.

For the proof, assume for a contradiction that *N* = *p* *q* is
a semi-prime such that *p*, *q* are distinct primes that have at
least four digits. Then the
condition that *N* is a ppn is equivalent to

*R*(*p* *q*) = *R*(1) + *R*(*p*) + *R*(*q*),

since the proper divisors of *N* are 1, *p*, *q*. That is to
say,

*R*(*p* *q*) = 1 + *R*(*p*) + *R*(*q*) (**).

Because the prime factors of *N* have at least four digits, none of them
can be = 2 or 5, so that *N* cannot be divisible by 10, hence cannot end
in the digit 0. Therefore, the number of digits of *N* is unchanged by
reversing *N*. Let *d*(*n*) denote the number of digits of *n*.
Recall that *d*(*n*) = _{}, where _{} denotes the greatest
integer __<__ x. Then_{}

*d*(*R*(*p* *q*)])

= *d*(*p* *q*)

= _{}

> _{}

= _{}

__>__ *d*(*p*) + *d*(*q*)
- 2

> max{*d*(*p*),*d*(*q*)} + 1 (since *p* and *q* each have at least four
digits)

__>__ *d*(1 + *R*(*p*) + *R*(*q*)).

Therefore, the left-hand side of (**) has more digits than the right-hand side
of (**), so that (**) cannot be satisfied. Hence, *N* cannot exist.

** **

**7. Picture-Amicable Pairs**

Let *D*(*n*) denote the
sum of *f*(*d*), where *d* ranges over all proper divisors of *n*.
For an arithmetical function *f*, a pair of numbers *a*, *b* is
called *f-amicable* if *f*(*b*) = *D*(*a*) and *f*(*a*)
= *D*(*b*) (cf. [P]). A pair *a*, *b* is called *picture-amicable*
(or *mirror-amicable*) if *a*, *b* is an *f*-amicable pair
for *f*(*n*) = *R*(*n*).

I had the pleasure of discovering the picture-amicable pair 2320000, 34049. The mirror equations of a picture-amicable pair can be defined similarly as for picture-perfect numbers. This pair has mirror equations

2320000
=* _{b}* 1 + 79 + 431

34049 =* _{b}*
1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 25 + 29 + 32 + 40 + 50 + 58 + 64 + 80 + 100
+ 116 + 125 + 128 + 145 + 160 + 200 + 232 + 250 + 290 + 320 + 40 + 464 + 500 +
580 + 625 + 640 + 725 + 800 + 928 + 1000 + 1160 + 1250 + 1450 + 1600 + 1856 +
2000 + 2320 + 2500 + 2900 + 3200 + 3625 + 3712 + 4000 + 4640 + 5000 + 5800 +
7250 + 8000 + 9280 + 10000 + 11600 + 14500 + 16000 + 18125 + 18560 + 20000 +
23200 + 29000 + 36250 + 40000 + 46400 + 58000 + 72500 + 80000 + 92800 + 116000
+ 145000 + 232000 + 290000 + 464000 + 580000 + 1160000.

The sum of the proper divisors of 34049 appear on the right side of the first equation, and the sum of the proper divisors of 2320000 appear on the right side of the second equation.

I used the following *Mathematica*
code to generate picture-amicable pairs {*n*, *b*}:

```
f[x_] :=
FromDigits[Reverse[IntegerDigits[x]]];
```

`d[x_] := Apply[ Plus, Map[ f, Drop[Divisors[ x], -1] ] ];`

`Do[a = d[n];`

`b = f[a];`

```
c = d[b]; u =
Last[IntegerDigits[a]];
```

```
If[u != 0 && n != b && c == f[n], Print[{n, b}]], {n, 2,
10^8}]
```

Readers may enjoy finding other picture-amicable
pairs or even picture-sociable chains.** **Since the reverse function is not
injective, my program does not perform an exhaustive search (not all possible
values of *b* are tested). Perhaps a resourceful reader will write an
exhaustive search program. This might appear to be an impossible task since
infinitely many b have to be inspected for a particular n. However, some
variation of Cantor’s “zigzag argument” for the countability of the rational
numbers can probably be used to circumvent this difficulty.

*Acknowledgments. *I would like to thank Jens Kruse
Andersen, Daniel Dockery, and Mark Ganson for their valuable contributions to
and interest in picture-perfect numbers; and Mohammad Khadivi for his
suggestions to improve this article.

[B] Beiler, A. “Recreations in the Theory of Numbers”, Dover Publications,

NY 1964.

[HW] Hardy, G. H. and Wright, E. M. “Introduction to the Theory of Numbers”

(Fifth Edition), Oxford University Press, 1983.

[P] Pe,
J. “On a Generalization of Perfect Numbers” in *The Journal of *

*
Recreational Mathematics *31(3).

[S]
Sloane, N. “The Online Encyclopedia of Integer Sequences”* *at the following

URL: http://www.research.att.com/~njas/sequences/Seis.html

[Su] Andersen, J., Dockery, D., Ganson, M., Pe, J. *Systematic
Undertaking to*

* Find
Picture-Perfect Numbers* (Super) Message Board at the following URL:

http://disc.server.com/Indices/187886.html

[W]
Wells, D. “The Penguin Dictionary of Curious and Interesting Numbers”,

Penguin Books, 1998.