Narcissistic number
The narcissistic numbers are a subset of natural numbers that generate themselves through certain arithmetic rules for their digits. However, they do not play a special role in pure mathematics, as they depend heavily on the number system used (usually the decimal system ) and therefore do not provide any real scientific benefit.
Armstrong numbers
An Armstrong number (according to Michael F. Armstrong) or PPDI (pluperfect digital invariant) is a narcissistic number, the sum of its digits, raised to the power of the number of digits, results in the number itself.
In other words:
An n-digit number of the form
- with and
is an Armstrong number if:
- .
Examples
Example 1:
An example of such a number with the power n = 5 is the five-digit number 54748:
Example 2:
The list of smallest narcissistic numbers with digits in the decimal system is the following (if there is no number with this number of digits, 0 is in this position):
- 1, 0, 153, 1634, 54748, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 0, 28116440335967, 0, 4338281769391370, 21897142587612075, 0, 1517841543307505039, 63105425988599693916, 128468643043731391252, 0, ... (sequence A014576 in OEIS )
There are a total of exactly 88 narcissistic numbers (excluding the 0) in the decimal system. The number of their positions indicates the following list of numbers:
- 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39 (episode A114904 in OEIS )
If you order these numbers according to their number of digits , you get the following table (sequence A005188 in OEIS ):
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|
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generalization
If you choose a different base , a narcissistic number is defined analogously to the decimal system:
An n-digit number with base b of the form
- with and
is a narcissistic number with base b if:
- .
Examples
Example 1:
The decimal is a narcissistic number with a base .
It is in four system (it is ), and actually applies to the then three-digit number: .
Example 2:
The decimal is a narcissistic number with a base .
It is in the six-system (it is ), and actually applies to the then five-digit number: .
A list of the narcissistic numbers with a base has already been given above (sequence A005188 in OEIS ).
The following is a list of the narcissistic numbers with a base , written in the respective system (where there are no further digits ) or in the decimal system:
Base b | narcissistic numbers to base b | narcissistic numbers to base 10 |
---|---|---|
2 | 0, 1 | 0, 1 |
3 | 0, 1, 2, 12, 22, 122 | 1, 2, 5, 8, 17 |
4th | 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303 (sequence A010343 in OEIS ) | 1, 2, 3, 28, 29, 35, 43, 55, 62, 83, 243 (episode A010344 in OEIS ) |
5 | 1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, 434434444, 1143204434402, 14421440424444 (sequence A010345 in OEIS ) | 1, 2, 3, 4, 13, 18, 28, 118, 289, 353, 419, 4890, 4891, 9113, 1874374, 338749352, 2415951874 (series A010346 in OEIS ) |
6th | 1, 2, 3, 4, 5, 243, 514, 14340, 14341, 14432, 23520, 23521, 44405, 435152, 5435254, 12222215, 555435035, 1053025020422, 1053122514003, 1435403205450, 1435403205451, 145524511445, 2535452502, 1450005114454 133024510545125, 13435022253535055, 15205355253553320, 15205355253553321, 105144341423554535 (episode A010347 in OEIS ) | 1, 2, 3, 4, 5, 99, 190, 2292, 2293, 2324, 3432, 3433, 6197, 36140, 269458, 391907, 10067135, 2510142206, 2511720147, 3866632806, 3866632807, 3930544834, 4953134588, 501864912975 124246559501, 4595333541803, 5341093125744, 5341093125745, 19418246235419 (episode A010348 in OEIS ) |
7th | 1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, 205145, 1424553, 1433554, 3126542, 4355653, 6515652, 125543055, 161340144, 254603255, 336133614, 542662326, ... (follow A010349 in OEIS ) | 1, 2, 3, 4, 5, 6, 10, 25, 32, 45, 133, 134, 152, 250, 3190, 3222, 3612, 3613, 4183, 9286, 35411, 191334, 193393, 376889, 535069, 794376, 8094840, 10883814, 16219922, 20496270, 32469576, 34403018, 416002778, 416352977, ... ( continuation A010350 in OEIS ) |
8th | 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, 40663, 42710, 42711, 60007, 62047, 636703, 3352072, 3352272, 3451473, 4217603, 7755336, 16450603, 63717005, 233173324, 3115653067, 4577203604, 61777450236, 147402312024, ... ( continuation A010351 in OEIS ) | 1, 2, 3, 4, 5, 6, 7, 20, 52, 92, 133, 307, 432, 433, 16819, 17864, 17865, 24583, 25639, 212419, 906298, 906426, 938811, 1122179, 2087646, 3821955, 13606405, 40695508, 423056951, 637339524, 6710775966, 13892162580, 32298119799, ... ( continuation A010354 in OEIS ) |
9 | 1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, 2225764, 6275850, 6275851, 12742452, 356614800, 356614801, 1033366170, 1033366171, 1455770342, 8463825582, 131057577510, 131057577511, ... ( continuation A010352 in OEIS ) | 1, 2, 3, 4, 5, 6, 7, 8, 41, 50, 126, 127, 468, 469, 1824, 8052, 8295, 9857, 1198372, 3357009, 3357010, 6287267, 156608073, 156608074, 403584750, 403584751, 586638974, 3302332571, 42256814922, 42256814923, 114842637961, ... (follow A010353 in OEIS ) |
10 | see above (sequence A005188 in OEIS ) | see above (sequence A005188 in OEIS ) |
... | ... | ... |
12 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 25, A5, 577, 668, A83, ... | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 125, 811, 944, 1539, 28733, 193084, 887690, 2536330, 6884751, 17116683, 5145662993, 25022977605, 39989277598, 294245206529, 301149802206, 394317605931, 429649124722, 446779986586, ... (Follow A161949 in OEIS ) |
... | ... | ... |
16 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, ... | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 342, 371, 520, 584, 645, 1189, 2458, 2729, 1456, 1457, 1547, 1611, 2240, 2241, 2755, 3240, 3689, 3744, 3745, 47314, 79225, 177922, 177954, 368764, 369788, 786656, 786657, 787680, 787681, 811239, 812263, ... ( continuation A161953 in OEIS ) |
... | ... | ... |
Example 3:
If you add up the k -th powers of the digits of a k -digit number n , you get (for n = 1, 2, 3, 4, ...) the following values:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53 , 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49, 50, 53, 58, 65, ... ( Follow A101337 in OEIS )
The above list can be interpreted as follows: for example, it says . Digit (this value is two-digit) the value . So if you raise the digits to the power of the number of their digits, that is , the result is . Indeed it is . If you get the exact value of the digit again (in this case it would have been) you would have found a narcissistic number.
properties
- The number of narcissistic numbers in a given base b is finite.
-
Proof:
- The maximum possible sum of k -th powers of a k -digit number in the base is . From a certain size of k , however, the following applies in any case . Thus, no narcissistic base number may have more than k digits, which means that there can only be a finite number of narcissistic numbers.
- Special case: Every narcissistic number in the decimal system must be less than .
-
Proof:
- Because of the above property must for k are -digit numbers: . This inequality has the solution .
- Thus, a narcissistic number in the decimal system cannot be greater than .
- There are only 88 narcissistic numbers in the decimal system. The largest narcissistic number in the decimal system has only 39 digits (instead of the maximum 60 digits given above) and is the following:
- All single digit numbers are narcissistic numbers (in any base).
- There is at least one two-digit narcissistic number in a base if and only if is not prime.
- The number of two-digit narcissistic numbers in the base is then , where is the number of positive divisors of (for example, is because 10 has the divisors 1,2,5 and 10).
- Any base that is not a multiple of has at least a three-digit narcissistic number. The bases without three-digit narcissistic numbers are the following:
- So there is no three-digit number with these bases .
Perfect digital invariant
A number, the sum of its digits, raised to the power of any number (and not its number of digits ), results in the number itself, is called a perfect digital invariant (or PDI ). These numbers are not narcissistic numbers , however . In contrast to the narcissistic numbers, there is no upper limit for the size of the number with PDIs (with a base ). It is also not known whether there are finitely or infinitely many PDIs given the basis .
Examples:
- The decimal number has four decimal places, but it can be represented as the sum of five powers of its decimal places:
- So it is a perfect digital invariant, but not a narcissistic number.
- The smallest PDIs to some power of their digits are
- The associated potencies are
- In the two upper lists (for example) the numbers 14459929 and 7 are in the 29th position. This means that the 8-digit number
- is.
- In the two upper lists, however, narcissistic numbers are also included. For example, in the 25th position are the numbers 1741725 and 7. This means that the 7-digit number is.
- The following list gives the smallest numbers that are equal to the sum of its digits with n th power ( n = 1, 2, 3, ...) (which 0 indicates that there is no such number):
- For example, it's in the sixth position . That means that is and that applies:
Narcissistic numbers with increasing potency
Narcissistic numbers with increasing potency are numbers whose sum of their digits, raised to the power of their place in the number (counting from the left), results in the number itself. For example a number abc = .
Examples:
- The following numbers are narcissistic in this sense:
Constant base narcissistic numbers
Constant base narcissistic numbers are numbers where the base is constant and the exponents are the digits of the number.
Example:
Wild narcissistic numbers
Wild narcissistic numbers are numbers where the way in which they generate themselves from their digits is not uniform.
Example:
Interesting numbers
Interesting numbers are even more free than the wild narcissistic numbers in their generation:
Examples:
See also
literature
- The Penguin Dictionary of Curious and Interesting Numbers . David Wells, ISBN 0-14-026149-4
Web links
- Eric W. Weisstein : Narcissistic Number . In: MathWorld (English).
- Check for narcissistic numbers in C # on .NET-Snippets.de
Individual evidence
- ^ Armstrong Numbers , Dik T. Winter
- ^ Lionel Deimel's Web Log
- ↑ PPDI (Armstrong) Numbers ( Memento of October 27, 2009 in the Internet Archive ), Harvey Heinz
- ↑ Thomas Jüstel: Special numbers. (PDF) Münster University of Applied Sciences , accessed on October 29, 2014 .
- ↑ Eric W. Weisstein : Narcissistic number . In: MathWorld (English).