Narcissistic Numbers
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.
Definition by Joseph S. Madachy, Mathematics on Vacation,
Thomas Nelson & Sons Ltd. 1966
He has lots of material on
narcissistic numbers on p 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+193195+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 = 3
3 + 44 + 33 +55438579088 = 4
4 + 33 + 88 +55 + 77 + 99 + 00 + 88 + 88These 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 = 4
5 + 82 + 66 +28 + 54397612 = 3
2 + 91 + 76 + 67 + 19 + 23The powers are the same as the digits, but in reverse order.
Thanks to Patrick De Geest for these two numbers
127 = -1 + 2
7 3125 = (31 + 2)5759375 = (7 - 5 + 9 - 3 + 7)5
Ascending & Descending Powers
1676 = 1
1 + 62 + 73 + 641676 = 1
5 + 64 + 73 + 62NOTE: the order of the powers.
43 = 42 + 33 63 = 62 + 33
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
89 = 81 + 92
Thanks to Patrick De Geest for these last two numbers
343 = (3 + 4)
3 3456 = 3! x 4/5 x 6!355 = 3 x 5
! - 5 4096 = (4 + 0 x 9)6715 = (7 - 1)
! - 5 5161 = 5! + (1 + 6)! + 1729 = (7 + 2)Ö
9 6859 = (6 + 8 + 5)Ö 9
Wild Narcissistic Numbers
24739 = 2
4 x 7! x 3923328 = 2 x 3
3! x 2 x 8For more Wild Narcissistic numbers see Mike Keith's home page at http://users.aol.com/s6sj7gt/mikewild.htm
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.
2
5·92 = 2592Here are some with fractions
1129 1/3 = 11
2·9 1/3 2124 9/11 = 212·4 9/11Some 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
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 = 2
70
and the sum of the digits in 270 equals 70.
| 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. |
|
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. |
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
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
666 = (61 + 61 + 61) + (63 + 63 + 63)
Also 666 = 16 - 26 + 36
Unique Factorials (Factorians)
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.
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?
4! + 1 = 52 5! + 1 = 112 7! + 1 = 712
Are there more of these numbers?
Clifford Pickover, Keys to Infinity, p. 170
The only solution for sum of subfactorials of digits ?
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
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
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.
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 They 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 leading to each PPDI, PDI or RDI of order-3
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..
|
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.
|
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, & Happy numbers
- 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
|
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 |
0 |
|
5 |
102 |
9 |
2, 2, 4, 6, 10, 10, 12, 22, 28 |
9 |
3 |
727 |
56 |
90 |
15578 |
3 |
|
6 |
50 |
5 |
2, 3, 4, 10, 30 |
7 |
1 |
300 |
91 |
1380 |
127889 |
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 |
maybe 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 |
| 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.