This article is about numbers with properties relating to their digits that I find interesting.
Pandigital numbers
A pandigital number is a number with all digits. This is not really that interesting, but there are some pandigital numbers with interesting properties:
123456789, the first zeroless pandigital number, has the following property: When you multiply the number by any single digit number not divisible by 3, you get a different permutation of the digits (type them all out on a numpad to see for yourself):
123456789 x 2 = 246913578
123456789 x 4 = 493827156
123456789 x 5 = 617283945
123456789 x 7 = 864197523
123456789 x 8 = 987654312
987654321 also has the same property if we assume that there is a leading zero before the 9:
(0)987654321 x 2 = 1975308642
(0)987654321 x 4 = 3950617284
(0)987654321 x 5 = 4938271605
(0)987654321 x 7 = 6913580247
(0)987654321 x 8 = 7901234568
All pandigital numbers with evenly distributed digits are divisible by 9, because 1+2+3+4+5+6+7+8+9 = 45, which is divisible by 9. 123456789 is equal to 9 * 13717421, and 987654321 is equal to 9*109739369. 1023456789 (the first true pandigital number), for example, is equal to 9 * 113717421, and 113717421 is itself equal to 9*12635269. Thus, a prime number with all digits must have at least one digit that appears more than once. The first pandigital prime is 10123457689.
In fact, in any even base (2b), a number with all digits distributed evenly will be divisible by 2b - 1. For example, 123456 = 186510 which is divisible by 5. And, 76543218 is equal to 2,054,353, which is 7 * 293,479.
Narcissistic numbers
Narcissistic numbers are numbers that are equal to the sum of their digits all raised to the power of the number of digits. All 1-digit numbers are narcissistic. There are no 2-digit narcissistic numbers in base 10 since 102 + 1 = 101, which is prime. There are only four numbers after 1 which are the sums of the cubes of their digits, those being 153, 370, 371, and 407.
There are also three 4-digit narcissistic numbers: 1634, 8208, and 9474. There are also three 5-digit narcissistic numbers: 54748, 92727, and 93084. After that, there is only one 6-digit narcissistic number: 548834. Then, there are four 7-digit narcissistic numbers: 1741725, 4210818, 9800817, and 9926315.
Eight 11-digit narcissistic numbers exist, the most narcissistic numbers of any number of digits greater than 1. They are 32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, and 94204591914.
There are no 12-digit, 13-digit, 15-digit, 18-digit, 22-digit, 26-digit, 28-digit, 30-digit, 36-digit, or 40-or-higher-digit narcissistic numbers.
The largest narcissistic number is 115,132,219,018,763,992,565,095,597,973,971,522,401, which also forms a pair with 115,132,219,018,763,992,565,095,597,973,971,522,400. It can be proven that there is a limit on these numbers since the sum of the n-th powers of the digits of an n-digit number can be no larger than n*9n, which has fewer than n digits for n > 62, and when approaching that limit, you would start requiring a lot of the digits to be 8s or 9s.
Ascending and descending power numbers
Ascending power numbers are numbers that are equal to the sums of the consecutive powers of their digits starting at the 1 power. Other than the trivial cases of the single-digit numbers, only 10 of these numbers exist. 89 is the only 2-digit ascending power number.
There are four 3-digit ascending power numbers: 135 (1^1 + 3^2 + 5^3), 175 (1^1 + 7^2 + 5^3), 518, and 598. Then, after that, there are only three 4-digit ascending power numbers: 1306 (1^1 + 3^2 + 0^3 + 6^4), 1676 (1^1 + 6^2 + 7^3 + 6^4), and 2427 (2^1 + 4^2 + 2^3 + 7^4). The next two ascending power numbers after that are the two largest ones, the 7-digit number 2,646,798, and the 20-digit number 12,157,692,622,039,623,539. The largest is just slightly more than 920 = 12,157,665,459,056,928,801 = 340.
The last digit of an ascending power number is usually the n-th root of the number where n is the number of digits in the number, rounded down. For instance, 2646798 is between 87 and 97 and ends with 8. The largest is between 920 and 1020 (much closer to the former), and ends with 9.
However, there is only one similar number for multiplication: 19, which is 1*1 + 9*2.
The definition of ascending power numbers can be extended to be any number that is equal to the sum of consecutive powers of its digits. With this definition, there are possibly infinitely many ascending power numbers. Below are the first ascending power numbers under this definition.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 43, 63, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539
43 and 63 are the only known numbers of the form abc...z = a2 + b3 + ... + zd+1 w/ d digits. There are none with 3, 4 or 5 digits.
Descending power numbers are numbers that are the sum of their digits each raised to consecutive powers going backwards. 1676 is also a descending power number because 1^5 + 6^4 + 7^3 + 6^2 = 1676. Two other descending power numbers are 24 (23 + 42) and 332 (35 + 34 + 23).
PDIs and PDDIs
A perfect digital invariant, sometimes abbreviated as a PDI, is a number that is equal to the sum of its digits each raised to some fixed power. This is the definition of narcissistic numbers with the requirement that the power be the number of digits loosened. For example,
4150 = 4^5 + 1^5 + 5^5 + 0^5
So the number 4150 is a perfect digital invariant, but not a narcissistic number. The next PDIs are 4151, 194979, and 14459929. 4151 and 194979 are both equal to the sums of the fifth powers of their digits, and 14459929 is equal to the sum of the seventh powers of its digits. The largest known PDI is the 41-digit number 36,428,594,490,313,158,783,584,452,532,870,892,261,556, which is equal to the sum of the 42nd powers of its digits.
A perfect digit-to-digit invariant, sometimes abbreviated to a PDDI, is a number that is equal to the sum of the digits each raised to the power of themselves. 3435 is a perfect digit-to-digit invariant because 3^3 + 4^4 + 3^3 + 5^5 = 3435. 438,579,088 is also a PDDI if 0^0 is interpreted to be equal to 0, but if 0^0 is undefined or 1, then 3435 is the only non-trivial PDDI.
There are no PDDIs other than 1 in bases 2, 5, and 8.
Constant base numbers
Constant base numbers are numbers that are equal to the sum of the numbers that are a certain base raised to the power of each of its digits. For example,
4624 = 4^4 + 4^6 + 4^2 + 4^4
The known constant base numbers are 1033 (8^1 + 8^0 + 8^3 + 8^3), 4624 (given above), 595968 (4^5 + 4^9 + 4^5 + 4^9 + 4^6 + 4^8), 3909511 (5^3 + 5^9 + 5^0 + 5^9 + 5^5 + 5^1 + 5^1), 13177388 (7^1 + 7^3 + 7^1 + 7^7 + 7^7 + 7^3 + 7^8 + 7^8), and 52135640 (19^5 + 19^2 + 19^1 + 19^3 + 19^5 + 19^6 + 19^4 + 19^0).
Below are the first several nontrivial palindromic squares. (Any palindromic square that is part of a pattern is trivial, except the smallest such case: for instance, 1001^2 = 1002001 is trivial, while 11^2 = 121 is not)
121
484
676
12321
14641
44944
69696
94249
698896
5221225
6948496
123454321
125686521
522808225
617323716
942060249
10221412201
1022325232201
1024348434201
1232346432321
1234567654321
9420645460249
102234363432201
123212464212321
123456787654321
12321024642012321
12343456865434321
12345678987654321
121242363484363242121
123432124686421234321
942064503484305460249
46501623417708833880771432610564
102212100000022262220000001212201
1635977102407987117897042017795361
10000022222212345854321222222000001
12100024202432104840123420242000121
18434129467955011411055976492143481
10000000222220001234743210002222200000001
9218713598451835185192915815381548953178129
123210000002220000002474200000022200000012321
14899578972945056149893218681239894165054927987599841