Narcissus, according to Greek mythology, fell in
love with his own image,
seen in a pool of water, and changed into the flower now called by his
name.
Since this section deals with numbers "in love with themselves",
narcissistic
numbers will be defined as those that are representable, in some way,
by
mathematically manipulating the digits of the numbers themselves.
Joseph S. Madachy, Mathematics on
Vacation, Thomas Nelson & Sons Ltd. 1966
There is lots of material on narcissistic numbers on pp 163 to 175 of this
book. |
CONTENTS
Constant Basenumbers
|
4624 = 44+46+42+44
1033 = 81+80+83+83 |
Note that the powers match the digits of the
number. |
|
595968 = 45+49+45+49+46+48
3909511 = 53+59+50+59+55+51+51
13177388 = 71+73+71+77+77+73+78+78
52135640 = 195+192+191+193+195+196+194+190
|
To the left are some of those sent to me by
Patrick de Geest in Dec.,1998, along with the name suggestion. |
3435
(PDDI's)
3435 = 33
+ 44
+ 33
+55
438579088 = 44
+ 33
+ 88
+55
+ 77
+ 99
+ 00
+ 88
+ 88
These are called Perfect Digit-to-Digit Invariants or
PDDI's for short. (See PDI, PPDI and
RDI at the bottom of this page.)
The only two integers in the decimal number system with this property (plus the
trivial 0 & 1).
D. Morrow ran a search up to 109 with no additional finds accept the
trivial adding of zeros to the above 2 numbers.
Note that here 00 is considered equal to 0. Normally 00 is
considered equal to 1 (see above 1033 where 80 = 1).
Curious and Interesting Numbers p.190
and D. Morrow JRM 27:1, 1995 p 9 and JRM 27:3, 1995, p205-207
Reverse of Above
48625 = 45
+ 82
+ 66
+28
+ 54
397612 = 32
+ 91
+ 76
+ 67
+ 19
+ 23
The powers are the same as the digits, but in reverse order.
Thanks to Patrick De Geest for
these two numbers
Noteworthy Numbers
127 = -1 + 27
3125 =
(31
+ 2)5
759375 = (7 - 5 + 9 - 3 + 7)5
Ascending &
Descending
1676 = 11
+ 62
+ 73
+ 64
1676 = 15
+ 64
+ 73
+ 62
NOTE: the order of the powers.
Ascending Powers
43 = 42
+ 33
63 = 62
+ 33
89
= 81
+
92
135 = 11
+ 32
+ 53
175 = 11
+ 72
+ 53
518 = 51
+ 12
+ 83
598 = 51
+ 92
+ 83
1306 = 11
+ 32
+ 03
+ 64
2427 = 21
+ 42
+ 23
+ 74
2646798 = 21
+ 62
+43
+ 64
+ 75
+ 96
+ 87
Thanks to Patrick De Geest for
these last two numbers
Interesting Numbers
343 = (3 + 4)3
3456 = 3!
x 4/5 x 6!
355 = 3 x 5!
- 5 4096 = (4 + 0 x 9)6
715 = (7 - 1)!
- 5 5161 = 5!
+ (1 + 6)!
+ 1
729 = (7 + 2)√9
6859 = (6 + 8 + 5)√9
Wild Narcissistic
Numbers
24739 = 24
+ 7!
+ 39
23328 =
2 x 33!
x 2 x 8
Printer’s errors
In 1917, H. E. Dudeney published a book of mathematical recreations
called Amusements in Mathematics. Amusement # 115 tells of a
printer when required to set the type for number 25•92,
mistakenly set it as 2592 (the dot was meant to indicate multiplication).
However, upon proofreading the number, it was found to be correct as
written.
The other numbers presented here were found by D. L. Vanderpool of
Pennsylvania and presented in J. S. Madachy's Mathematics on Vacation,
1966, 17-147099-0.
25·92
= 2592
Here are some with fractions
1129 1/3 = 112·9
1/3 2124 9/11 = 212·4
9/11
Some lead to infinite series of errors
34425 = 34·425
312325 = 312·325
344250 = 34·4250
3123250 = 312·3250
3442500 = 34·42500
31232500 = 312·32500
etc etc
Sums & Powers
81
= (8+1)2
= 92
512 = (5+1+2)3
= 83
4913 = (4+9+1+3)3
= 173
17576 = (1+7+5+7+6)3
= 263
234256 = (2+3+4+2+5+6)4
= 224
1679616 = (1+6+7+9+6+1+6)4
= 364
17210368 = (1+7+2+1+0+3+6+8)5
= 285
205962976 = (2+0+5+9+6+2+9+7+6)5
= 465
8303765625 = (8+3+0+3+7+6+5+6+2+5)6
=
456
24794911296 = (2+4+7+9+4+9+1+1+2+9+6)6
= 546
271818611107 = (2+7+1+8+1+8+6+1+1+1+0+7)7
= 437
6722988818432 = (6+7+2+2+9+8+8+8+1+8+4+3+2)7
= 687
72301961339136 =
(7+2+3+0+1+9+6+1+3+3+9+1+3+6)8
= 548
248155780267521 =
(2+4+8+1+5+5+7+8+0+2+6+7+5+2+1)8
= 638
Where a digital invariant was defined as a number equal to the sum of
the nth powers of its digits, this category has numbers equal to a power of
the sums of their digits.
J.S.Madachy, Mathematics On Vacation p.167 - 170 presents an
algorithm that results in a relatively small search field for numbers of this
type. It turns out there are 432 such numbers in the range to P101 ,
the largest, having 320 digits with a digit sum of 1468, is 1468101.
A related number
1,180,591,620,717,411,303,424
= 270
and the sum of the digits in
270 equals
70.
Sums of 2 squares
12 33 = 122
+ 332
990 100 = 9902
+ 1002
9412
2353 = 94122
+ 23532
74160 43776 =
741602
+ 437762
116788 321168 = 1167882
+ 3211682 |
Each number is equal
to the sum of the squares of its two halves. 88 33 = 882
+ 332
(supplied Oct./09 by
Patrick Vennebush)
Is this the only other 4 digit number with
this property? |
4 8 = 82
- 42
34 68 = 682
- 342
416 768 = 7682
- 4162
3334 6668 = 66682
- 33342 |
Each number is equal
to the difference of the squares of its two halves.
Does a pattern like this exist for
sum of the squares of its two halves? |
22 18 59 = 223
+ 183
+ 593
166 500 333 = 1663
+ 5003
+ 3333 |
Each number is equal to the sum of the cubes of its
three thirds. |
Amicable pairs
3869 = 622
+ 052
and
6205 = 382
+ 692
5965 = 772
+ 062
and
7706 = 592
+ 652
Each number of the pair is
equal to the sum of the squares of the two halves of the other number.
And somewhat similar
13+33+63
= 244 and 23+43+43
= 136
Power Sum Numbers
298 = (22
+ 92
+ 82)
+ (22
+ 92
+ 82)
336 = (31
+ 31
+ 61)
+ (32
+ 32
+ 62)
+ (33
+ 33
+ 63)
444 = (41
+ 41
+ 41)
+ (42
+ 42
+ 42)
+ (43
+ 43
+ 43)
+ (43
+ 43
+ 43)
Above are examples of power-sum numbers.
The number 336 is a subclass called proper because the groups of
exponents are all distinct.
M. Keith, Journal of Recreational
Mathematics 18:4 1985-86, p 275
Power Sum 666
666 = (61
+ 61
+ 61)
+ (63
+ 63
+ 63)
Also
666 = 16
- 26
+ 36
Unique Factorials
1 = 1!
2 = 2!
145 = 1!
+ 4!
+ 5!
40585 = 4!
+ 0!
+ 5!
+ 8!
+ 5!
These are the only integers with this
property. Remember factorial 0 is 1 by definition.
Clifford Pickover calls these numbers Factorians. See his Keys to Infinity,
p.169-171.
Factorial Products
0!
* 1!
= 1!
1!
* 2!
= 2!
6!
* 7!
= 10!
1!
* 3!
* 5!
= 6!
1!
* 3!
* 5!
* 7!
= 10!
Are these the only examples of factorials that
are the products of factorials in arithmetic sequence or progression?
Brown Numbers
4! + 1 = 52
5! + 1 = 112
7! + 1 = 712
Are there more of these numbers?
Clifford Pickover, Keys to
Infinity, p. 170
Sum of subfactorials
148,349 =
!1 +
!4
+ !8
+ !3
+ !4
+ !9
The exclamation point in front of the number
indicates it is a sub-factorial.
Subfactorials are defined as follows:
The subfactorials of the digits are : !0 = 0,
!1 = 0, !2 = 1, !3 = 2, !4 = 9, !5 = 44, !6 = 265, !7 = 1854, !8 = 14833, !9 =
133496.
J. S. Madachy, Mathematics on
Vacation, p. 167
Perfect Digital
Invariants
4150 = 45
+ 15
+ 55
+ 05
4151 = 45
+ 15
+ 55
+ 15
194979 = 15
+ 95
+ 45
+ 95
+ 75
+ 95
14459929 = 17
+ 47
+ 47
+ 57
+ 97
+ 97
+ 27
+ 97
A PDI is a number equal to the sum of
a power of its digits when the power is not equal to the length of the number.
A 41 digit PDI is
36,428,594,490,313,158,783,584,452,532,870,892,261,556.
It is equal to the sum of each of its digits raised to the 42nd
power.
L. E. Deimel, Jr and M. T.
Jones, JRM,14:4, 1981-82 p284
PPDI (Armstrong)
Numbers
153 = 13
+ 53
+ 33
1634 = 14
+ 64
+ 34
+ 44
54748 = 55
+ 45
+ 75
+ 45
+ 85
548834 = 56
+ 46
+ 86
+ 86
+ 36
+ 46
1741725 = 17
+ 77
+ 47
+ 17
+ 77
+ 27
+ 57
24678050 = 28
+ 48
+ 68
+ 78
+ 88
+ 08
+58
+ 08
146511208 = 19
+ 49
+ 69
+ 59
+ 19
+ 19
+ 29
+ 09
+ 89
4679307774 = 410
+ 610
+ 710
+ 910
+ 310
+ 010
+ 710
+ 710
+ 710
+ 410
82693916578 = 811
+ 211
+ 611
+ 911
+ 311
+ 911
+ 111
+ 611
+ 511
+ 711
+ 811
The above numbers are called Pluperfect
Digital Invariants or PPDIs. They are also called Armstrong Numbers. In each
case, the power corresponds to the number of digits.
There are no PPDIs for numbers of 2, 12 or 13
digits. The number shown for 11 digits is one of eight such numbers.
Largest possible PPDI has 39 digits. It is
115,132,219,018,763992,565,095,597,973,971,522,401.
It is equal to the sum of the 39th power of its digits.
NOTE that all single digit numbers, in all
bases, are PPDIs. The other comments above refer to base 10 PPDIs.
See L. Deimel, Jr. & M. Jones,
Finding Pluperfect Digital Invariants, JRM vol. 14:2, 1981-82, p 87-107
for a list of PPDI's in number bases 2 to 10, in base ten all 88 PPDI's to
order-39. Also 6 references.
Lionel Deimel has a web page on
Armstrong Numbers here.
Recurring
Digital Invariant
Each number of
each of the following two series is known as a Recurring Digital
Invariant or RDI.
Here is an order
three RDI, 55, with two intermediate numbers before 55 appears again. The order
four RDI, 1138, has six intermediate numbers before 1138 reappears.
Notice that RDI’s are not necessarily Armstrong numbers i.e. the power is not
necessarily the same as the length of the number.
RDI’s,
PDI's
and PPDI’s
are members of a larger class of numbers called narcissistic. A
narcissistic number is defined as one that may be represented by some
manipulation of its digits.
55 : 53
+ 53
= 250
250 : 23
+ 53
+ 03
= 133
133 : 13
+ 33
+ 33
= 55 |
This is one of four RDI cycles of order-3
The others are:
136, 244 length 2
919, 1459 length 2
55, 250, 133 length 3 ( the one to the left)
160, 217, 352 length 3
The four PPDI's: 153, 370, 371, 407 may each
be considered a cycle of length 1. |
1138 : 14
+ 14
+ 34
+ 84
= 4179
4179 : 44
+ 14 + 74
+ 94 = 9219
9219 : 94
+ 24 + 14
+ 94 = 13139
13139 : 14
+ 34 + 14
+ 34 + 94
= 6725
6725 : 64
+ 74 + 24
+ 54 = 4338
4338 : 44
+ 34 + 34
+ 84 = 4514
4514 : 44
+ 54 + 14
+ 44 = 1138 |
There is one other order-4 RDI. It has a cycle length of
two and consists of 2178 and 6514. Also the three
order 4 PPDI's
1634, 8208, 9474 may be considered cycles of length one.
|
Example
strings to PPDI
The sum of the cubes of the digits of
each number forms the next number in the string until a cycle of length 1,
2 or 3 is reached..
Happy Number
7
72
= 49
42
+ 92
= 97
92
+ 72
= 130
12
+ 32
+ 02
= 10
12
+ 02
= 1 |
1
is the first happy number.
7 is the
second happy number.
Iterating the process
of summing the square of the decimal digits of a number, you either
reach ---the RDI cycle 4, 16,
37, 58, 89 145, 42, 20 and
back to 4
or ---you reach the number
1.
Starting number's that
end up at number 1 are called Happy Numbers. |
Happy number 7 requires 5 iterations before it
reaches the number 1.
Happy number 356 requires 6 iterations before it reaches the number 1.
Happy number 78999 requires 7 iterations before it reaches the number 1.
The 10,012,125th Happy number is 71,406,333 and at this point 7 is still the
maximum iterations required.
Is 7 the maximum iterations required
for any number to evolve to 1 when each digit is squared and then summed?
Defined by R. K. Guy, Unsolved
Problems in Number Theory, Springer-Verlag.
Summary: PDI, PPDI,
RDI
-
Raise each
digit of any
number to any
power and then sum to make a new number.
-
Repeat these
steps for this new number (but use the same power used for the original
number). Eventually, you will
enter a closed loop where the numbers generated repeat indefinitely.
-
If the loop
is of length one you have reached a
PDI
if the power each digit is raised to is different then the length of the
number.
If the power used is the same as the length of the number, this number
is a PPDI.
If the number reached is one, and you have been raising each digit by
the power 2,
the starting number is a Happy
number.
-
If the length
of the loop is greater then one, this is an
RDI.
The RDI is always one of those you would have obtained if the original
starting number had been the same length as the power used.
Example; say the starting number is 12345 and each digit is raised to
the third power. The RDI eventually reached will be the same as
if the starting number was of length three.In this case after 9
iterations we will reach the PPDI 153.This can be considered as an RDI
cycle of length one.
The order of the RDI
is always the same as the
power each digit is raised to before summing.
-
The largest
possible PPDI
(in base 10) consists of 39 digits.
There is no such restriction on PDI's.
A 41 digit
PDI
is 36,428,594,490,313,158,783,584,452,532,870,892,261,556.
It is equal to the sum of each of its digits raised to the 42nd
power.
L. E. Deimel, Jr and M. T. Jones, JRM,14:4,
1981-82 p284
Summary Table of
PPDI's
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
Ord-er # |
# of entry
points |
# of RDI Cyc. |
What RDI Cycles |
Empty entry
points |
# of PPDIs |
Entries to PPDIs |
maximum
Iterations required |
# of starting
numbers |
Root start # |
PDI's |
|
2 |
9 |
1 |
8 |
1 |
0 |
0 |
9 |
1 |
60 |
0 |
|
3 |
15 |
4 |
2, 2, 3, 3 |
0 |
4 |
761 |
13 |
3 |
177 |
0 |
|
4 |
12 |
2 |
2, 7 |
1 |
3 |
643 |
52 |
18 |
2899
5599 |
0 |
|
5 |
102 |
9 |
2, 2, 4, 6, 10, 10, 12, 22, 28 |
9 |
3 |
727 |
56 |
90 |
15578
37799 |
3 |
|
6 |
50 |
5 |
2, 3, 4, 10, 30 |
7 |
1 |
300 |
91 |
1380 |
127889
205678
227899
445799
455566 |
0 |
|
7 |
267 |
11 |
2, 2, 3, 6, 12, 14, 21, 27, 30, 56,
92 |
18 |
4 |
28140 |
106 |
360 |
3055588 |
1 |
|
8 |
182 |
3 |
3, 25, 154 |
|
3 |
|
There may be a bigger
string |
|
9 |
299 |
13 |
2, 3, 3, 4, 8, 10, 10, 19, 24, 28,
30, 80, 93 |
|
4 |
|
|
|
|
|
|
10 |
234 |
6 |
2, 6, 7, 17, 81, 123 |
|
1 |
|
|
|
|
|
|
11 |
539 |
9 |
5, 7, 18, 20, 42, 48, 117, 118, 181 |
|
8 |
|
|
|
|
|
|
12 |
267 |
3 |
40, 94, 133 |
|
0 |
|
|
|
|
|
|
13 |
297 |
6 |
5, 8, 16, 22, 100, 146, |
|
0 |
|
|
|
|
1 |
|
14 |
571 |
5 |
14, 15, 65, 96, 381 |
|
1 |
|
|
|
|
|
|
15 |
829 |
7 |
8, 12, 30, 46, 75, 216, 362 |
|
0 |
|
|
|
|
|
Column Notes:
|
1 |
Order of the PPDI i.e. the
power each digit of the number is raised to. Also the length of each
starting number. Each number in this range is evaluated except for the first
one. 10, 100, 1000, etc always converges to the number 1. |
|
2 |
These entry points are the
value all numbers in the range must eventually reduce to. They are PPDI's,
PDI's, or members of an RDI cycle.
NOTE: I show only PDI's that have numbers in this range converging to them.
There may be some PDI's in this range that are a result of numbers from
other orders. All members of an RDI cycle are represented here although some
may have NO numbers converging to them. |
|
3 |
Number of individual RDI
cycles. Cycles of length 1 (PPDI's and RDI's) are not shown here. |
|
4 |
Lists the actual cycle
lengths. There is often more then 1 cycle of the same length. |
|
5 |
Some numbers in an RDI cycle
may not have any numbers reducing to them, i.e this is NOT an entry point to
the cycle. |
|
6 |
Actual number of PPDI's of
this order. Each one is of course one number in the range, but other numbers
may also reduce to it after several or many iterations. |
|
7 |
Shows how many numbers reduce
to the PPDI's. |
|
8 |
The maximum iterations
required of a number in the range before it becomes a PPDI, a PDI (or 1), or
a member of an RDI cycle. |
|
9 |
The number of starting
numbers requiring the maximum iterations. |
|
10 |
The starting number for
maximum iterations. All numbers with permutations of these digits are also
start numbers. |
|
11 |
Number of PDI's. Each of
these is reached eventually from some number in the range. The first number
in every range (10, 100, 1000, etc) always generates the number one. This
fact is not shown in this column. Only orders 2 and 3 have other numbers
that converge to one. |
NOTE: Number of
RDI cycles for orders 10 – 15 may not be accurate (I may have missed some).
Can you add to this table?
I have a word
document containing more detailed notes resulting from this investigation. Is is
called PPDI.doc (236 kb) and may be downloaded from my
download page.
|
