By

Joseph L. Pe

iDEN System Engineering Tools and Statistics

Motorola

21440 West Lake Cook Road

Deer Park, IL 60010

Email: ajp070@email.mot.com

ABSTRACT. This paper
discusses three of the author’s digit-reversal recreations. First,
picture-perfect numbers are numbers n such that the reverse of n equals the sum
of the reverses of the proper divisors of n. Second, the palinpoints of an
arithmetical function f are arguments at which f commutes with the reverse
function. Third, the fixated points of f are arguments n such that n and
reverse(n) appear in f(n). This paper also mentions several conjectures and
open problems, and includes source code to start the reader on his own computer
explorations.

**1. ****Preliminaries**

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 paper
will also be integer-valued and have the set of natural numbers as its domain.

**2.
****Perfect number recreations**

** **

In [P], I 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 some perfect number sequences), it can also provide amusing and curious relations for f that transforms its argument’s digits in some way.

For 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.

In the next section, the perfect numbers resulting from f(n) = reverse(n) (e.g., f(123) = 321) 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 (that is, its divisors excluding itself). 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 backwards gives

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 *0*s.) In
short, n is an f-perfect number, where f(n) = reverse(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 is 6, but since 6 and its proper divisors are single digit, it is trivially picture-perfect. The first non-trivial picture-perfect number is 10311.

If P is a
picture-perfect number, then call the expression P =* _{b}* D, where D is the (unevaluated) sum of the
proper divisors of P, the

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

Here is *Mathematica*
code to generate picture-perfect numbers not exceeding 10^{8}:

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

`Do[If[f[n]==Apply[Plus,Map[f,Drop[Divisors[n],-1]]],Print[n]],{n,2,10^8}]`

Some versions of *Mathematica*
will not allow
the Do loop iterators to be larger
than 10^{10}, but this code using a While loop does not suffer from
this drawback:

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

n
= 2; (*Place initial value of n here*)

While[n<10^12, `If[f[n]==Apply[Plus,Map[f,`

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

When my computer
search had turned up no new picture-perfect numbers 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 picture perfect number: 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 picture-perfect number raised hopes that the set of such numbers is infinite, and that a fourth number would be found before long. Two puzzle enthusiasts, Daniel Dockery and Mark Ganson, soon joined me in the search. Our team has a discussion forum at [Su], and membership is open to the interested public. Ganson provides free a Windows search utility he has written ([Su]).

We focused our efforts on the interval from 10^{9}
to 10^{10}. Three weeks later, Dockery [D] found the fourth picture-perfect
number. 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, our progress was
slow, even though Ganson’s C++ search program ran at least twice faster than *Mathematica*.
Then Jens Kruse Andersen ([An]) joined the team by announcing his discovery of
a new picture-perfect number: 7980062073 with mirror equation

7980062073 =_{b} 1 + 3 + 19 + 57 + 140001089 +
420003267 + 260020691.

Devising an efficient algorithm
that caches divisor information, Andersen quickly tested all numbers up to 10^{10},
and concluded that there are no more picture-perfect numbers in this range
other than the five already mentioned above.

Currently, our team is exhaustively
searching the interval from 10^{10 }to 10^{12 }using Andersen’s
program; see [Su] for progress updates and program download.

The sequence

6, 10311, 21661371, 1460501511, 7980062073 …

is sequence A069942 in [S]. “Small”
picture-perfect numbers are extremely rare pearls in the infinite ocean of
numbers—there are only five picture-perfect numbers below 10^{10}.
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 making (2^{n}-1)**
**a prime. Is there an expression yielding (not necessarily all)
picture-perfect numbers?

As this paper was nearing its final
draft, Andersen discovered the remarkable result that if the decimal number p =
140z10n89 is prime, then the product 57p is picture-perfect, and conversely,
where z is any number (possibly none) of *0*s and n is any number
(possibly none) of *9*s. In particular, if n has no *9*s, then 57p
has the form 798z62073; if n has at least one *9*, then 57p has the form
798z626m373 where m has one *9* less than n. The proof of Andersen’s
beautiful theorem appears towards the end of this section.

Call a prime p of the form
140z10n89 an *Andersen prime*, and its corresponding picture-perfect
number 57p 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**. Ganson [Ga] has used Andersen’s
theorem to identify 204 (mostly huge) Andersen numbers, the largest with 177
digits! This is the gargantuan 798z626m373 where z has 77 *0*s and m has 91 *9*s.

The sequence of picture-perfect numbers is an intriguing example of a sequence that appears at first to be extremely sparse—even finite—but yields several terms in the scale of the very large. An open problem is whether picture-perfect numbers such as 10311 can generate other picture-perfect numbers in the same way that 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 140z10n89, then reverse(57p) = 170 + reverse(p) + reverse(3p) + reverse(19p).

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

1, 3, 19, 57, p, 3p, 19p.

Adding the reverses of these proper divisors,

reverse(1) + reverse(3) + reverse(19) + reverse(57) + reverse(p) + reverse(3p) + reverse(19p)

= 170 + reverse(p) + reverse(3p) + reverse(19p)

= reverse(57p), by the lemma.

Hence, 57p is picture-perfect.

Conversely, if p is not prime, then 57 p will have more divisors, and the sum of the reversed divisors becomes larger than 170 + reverse(p) + reverse(3p) + reverse(19p) = reverse(57p). Hence, 57p 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*, hence, p = 140z10n89. In what follows, the finite sequences
z, n, and m (as defined above) appear in **boldface**. To simplify notation,
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**.

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

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

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

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

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

(The reader should
do the multiplications by hand to better understand the numerical patterns at
work here. Of course, the proof can be written even more formally—and more
opaquely—by stating that 57p = 798**z**626**m**373, 19p = 266**z**208**m**791,
3p = 420**z**32**n**67, and so on.)

Adding reverse(19p), reverse(3p), reverse(p), and 170 gives reverse(57p), 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 *9*s,
hence p = 140z1089. The proof here is handled similarly as in case (1). I only
mention the final summation of reverse(19p), reverse(3p), reverse(p), and 170
to reverse(57p), using the same notation as case (1):

19602**00**662

+
7623**00**024

9801**00**041

__ 170__

37026**00**897

**5.
****Observations and Conjectures**

** **

** **Andersen
conjectures that there are infinitely many Andersen primes, 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. Equivalently, the digital sum of each picture-perfect number is 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. Andersen asserts the weak result that every picture-perfect number has a prime factor less than 100 ([Su]).

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.

_{ }

Ganson observed that the digital sum of each of these numbers is divisible by 5 and conjectures that this is true for all base-6 picture-perfect numbers. (Compare this to his base-10 conjecture.)

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 p is (trivially) picture-perfect in any base b larger than p. This is because p and its proper divisors have one-digit base-b representations. For example, 6 and 28 are picture-perfect in base 29.

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}]

**6. 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) = reverse(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.

**7. Palinpoints of
arithmetical functions**

311^{2}
= 76721

113^{2}
= 12767

(where 113 and 12767 in the second equation are reverses of 311 and 76721, respectively, in the first equation) can be compactly expressed by the single equation

f(reverse(311)) = reverse(f(311)).

To generalize, let f be an
arithmetical function. The *palinpoints*
of f are arguments at which f commutes with the reverse function, that is,

f(reverse(n)) = reverse(f(n)).

As usual, ignore leading *0*s.

Hence, 311 is a palinpoint of f(n)
= n^{2}. (Martin Gardner in [Ga], p. 97 mentions that 13^{2 }=
169 and 31^{2 }= 961, but the notion of palinpoint for arbitrary
arithmetical functions in this paper appears to be new.)

As examples, let φ(n) and σ(n) denote Euler’s totient function and the sum-of-divisors function, respectively. The sequence of palinpoints of φ(n) begins

1, 2, 3, 4, 5, 6, 7, 8, 9, 535, 767, 2110, 6188, 6991, 8299, 8816, 9928...

(A069282
in [S].) For example,

φ(9928)
= 4608

φ(8299)
= 8064.

The sequence of palinpoints of σ(n) begins

1, 2, 3, 4, 5, 7, 14, 41, 124, 194, 333,
421, 491, 1324, 4231…

(A069514
in [S]).

Now,
let p(n) denote the n-th prime number. The first seven palinpoints of f are 1,
2, 3, 4, 5, 12, 21 (A069469 in [S]). The first four of these are trivial palinpoints
since p(1), …, p(4) are one-digit numbers. In the non-trivial n = 21, for
example, p(21) = 73 and p(12) = 37. However, p has no palinpoints between 21
and 10^{6} (which is as far as I have searched). The reader may find it
a challenging recreation to search for larger palinpoints of p, or prove that
no others exist.

Here
is *Mathematica *code to find palinpoints of p below 10^{8}:

`rev[n_] := FromDigits[Reverse[IntegerDigits[n]]];`

`f[n_] := Prime[n];`

```
Select[Range[10^8], f[rev[#]] == rev[f[#]]
&]
```

Returning
to f(n) = n^{2}, the palinpoints of f not exceeding 1000 are

1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 30,
31, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 130, 200, 201, 202,
210, 211, 212, 220, 221, 300, 301, 310, 311, 1000

I
attempted to submit this sequence to the Online Encyclopedia of Integer
Sequences ([S])—or OEIS for short—and was surprised to find that it was already
there, albeit in a different guise. In the OEIS this was characterized as the
sequence of numbers n such that

(sum of digits of n)^{2} = sum of
digits of n^{2 }

(A061909
in [S]). I conjectured that the palinpoint sequence and the OEIS sequence are
identical.

Robert
Israel [I] of the Department of Mathematics, University of British Columbia,
provided a partial answer to this conjecture. He proved that every term of the
OEIS sequence is in the palinpoint sequence. His proof now follows.

First,
we need a lemma: n is in the OEIS sequence if and only if there are no carries
in calculating n^{2}. For let d_{k}(n) be the digit of n in the
10^{k} ‘s place, so that

x
__311__
x __113__

311 339

311 113

__933____ __113____

Is the converse true, that is to
say, is every n in the palinpoint sequence also in the OEIS sequence?
Equivalently, is the calculation of n^{2} without carries? This
question is still unanswered. Israel has verified that all palinpoints not
exceeding 9 x 10^{5} are in the OEIS sequence. However, he observes
that several palinpoints have carries in bases other than 10. In binary, for
example, 3 = 11_{2} is a base-2 palinpoint since it is a base-2
palindrome, but the base-2 digital sum of 3^{2 }= 9 = 1001_{2}
is not equal to the square of the base-2 digital sum of 3. Examples in other
bases include:

1478 = 2000202_{3}

32 = 44_{7}

195 = 303_{8}.

** **

**8. Fixated points of
arithmetical functions**

Most
functions do not have any fixed points, and many have only one such point.
(Recall that x is a *fixed point* of a function f if f(x) = x.)

The
situation appears to be much rosier for "fixated points" of
arithmetical functions. n is a *fixated point* of an arithmetical function
f if, treated as strings, n and reverse(n) are both sub-strings of f(n).

The
notion of fixated points can be seen as a generalization of palindronomic
numbers. This is because the palindronomic numbers are exactly the fixated
points of the identity function id(n) = n which do not end in the digit *0*
(Those that end in *0* lose the *0* when reversed since leading *0*'s
are ignored. E.g., 1410 is not palindronomic, but is a fixated point of id(n).)

For
example, consider f(n) = 2^{n}. Then n = 49 is a fixated point of f
since f(49) = 2^{49} = 562**949**953421312 in which both 49 and its
reversal 94 appear as sub-strings. The sequence of fixated points of f begins

6, 10, 44, 49, 60, 67, 190, 191, 226,
252, 321, 373...

(A068588
in [S].) To take another example, let f(n) = Fibonacci(n), the n-th Fibonacci
number. (Recall that f is defined recursively by f(1) = f(2) =1, f(n) = f(n-1)
+ f(n-2) for n > 2.) Then n = 43 is a fixated point of f since f(43) =
Fibonacci(43) = **4334**94437 in which both 43 and its reversal 34 appear as
sub-strings. The sequence of fixated points of f begins

1, 5, 41, 43, 65, 70, 85, 99, 101, 194,
214, 340, 368...

The
reader can verify as an exercise that the sequence of fixated points of f(n) =
n^{3} has first few terms

1, 4, 5, 6, 9, 10, 40, 50, 60, 90, 99,
100, 400, 500...

and
the sequence of fixated points of f(n) = n^{4} begins

1, 5, 6, 10, 50, 60, 92, 100, 363, 500,
600, 636, 909, 1000...

As
a rule of thumb, the faster f(n) grows with respect to n, the more digits f(n)
will have as compared with n, hence, the likelier it is to contain n and
reverse(n) as sub-strings. Thus, it is more interesting to consider f(n) which
grows slowly with respect to n. One such function is given by f(n) = p(n), the
n-th prime, which has a rate of growth with respect to n that is approximately
equal to ln(n). The only fixated points of f not exceeding 10^{7} are 7
and 6460. p(7) = 17, which contains both 7 and its reversal 7; and p(6460) =
64601, which contains both 6460 and its reversal 646 (= 0646). Can the reader
find any more such points?

To
aid in the search, here is *Mathematica* code to generate fixated points:

`r = {};`

```
Do[m
= Prime[n];
```

```
s
= ToString[m];
```

```
If[StringPosition[s,
ToString[n]] != {} &&
```

```
StringPosition[s,
ToString[FromDigits[Reverse[IntegerDigits[n]]]]] != {},
```

```
r
= Append[r, n]], {n, 1, 10^10}];
```

```
r
```

To
use this code for a different function, just replace the "m =
Prime[n]" by the appropriate expression.

*Acknowledgments. *I would like to thank Jens Kruse
Andersen, Daniel Dockery, Mark Ganson, and Robert Israel for their valuable
contributions to and interest in these math recreations; and Mohammad Khadivi
for his suggestions to improve this paper.

[An]
Andersen, J. K. *Personal Communication.* 2002. (Appears in [Su])

[A] Andrews, G. “Number Theory”, Dover Publications, NY 1994.

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

NY 1964.

[D]
Dockery, D. *Personal Communication*. 2002. (Appears in [Su])

[G]
Ganson, M. *Personal Communication*. 2002. (Appears in [Su])

[Ga] Gardner, M. “The Magic Numbers of Dr. Matrix”, Prometheus Books, 1985.

[I]
Israel, R. *Personal Communication*. 2002.

[P] Pe,
J. “On a Generalization of Perfect Numbers”, to appear 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] *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.