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

${\displaystyle a=a_{n}\cdot 10^{n-1}+a_{n-1}\cdot 10^{n-2}+a_{n-2}\cdot 10^{n-3}+\ldots +a_{2}\cdot 10^{1}+a_{1}\cdot 10^{0}}$with and ${\displaystyle 0\leq a_{i}\leq 9}$ ${\displaystyle 1\leq i\leq n}$

is an Armstrong number if:

${\displaystyle a_{n}^{n}+a_{n-1}^{n}+a_{n-2}^{n}+\cdots +a_{1}^{n}=a}$.

Examples

Example 1:

An example of such a number with the power n = 5 is the five-digit number 54748:

${\displaystyle 54748=5^{5}+4^{5}+7^{5}+4^{5}+8^{5}=3125+1024+16807+1024+32768}$

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): ${\displaystyle n}$

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 ): ${\displaystyle n}$

n narcissistic numbers to base 10 1 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 3 153, 370, 371, 407 4th 1634, 8208, 9474 5 54748, 92727, 93084 6th 548834 7th 1741725, 4210818, 9800817, 9926315 8th 24678050, 24678051, 88593477 9 146511208, 472335975, 534494836, 912985153 10 4679307774 11 32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, 94204591914 14th 28116440335967 16 4338281769391370, 4338281769391371

n narcissistic numbers to base 10 17th 21897142587612075, 35641594208964132, 35875699062250035 19th 1517841543307505039, 3289582984443187032, 4498128791164624869, 4929273885928088826 20th 63105425988599693916 21st 128468643043731391252, 449177399146038697307 23 21887696841122916288858, 27879694893054074471405, 27907865009977052567814, 28361281321319229463398, 35452590104031691935943 24 174088005938065293023722, 188451485447897896036875, 239313664430041569350093 25th 1550475334214501539088894, 1553242162893771850669378, 3706907995955475988644380, 3706907995955475988644381, 4422095118095899619457938

n narcissistic numbers to base 10 27 121204998563613372405438066, 121270696006801314328439376, 128851796696487777842012787, 174650464499531377631639254, 177265453171792792366489765 29 14607640612971980372614873089, 19008174136254279995012734740, 19008174136254279995012734741, 23866716435523975980390369295 31 1145037275765491025924292050346, 1927890457142960697580636236639, 2309092682616190307509695338915 32 17333509997782249308725103962772 33 186709961001538790100634132976990, 186709961001538790100634132976991 34 1122763285329372541592822900204593 35 12639369517103790328947807201478392, 12679937780272278566303885594196922 37 1219167219625434121569735803609966019 38 12815792078366059955099770545296129367 39 115132219018763992565095597973971522400, 115132219018763992565095597973971522401

generalization

If you choose a different base , a narcissistic number is defined analogously to the decimal system: ${\displaystyle b\neq 10}$

An n-digit number with base b of the form

${\displaystyle a=a_{n}\cdot b^{n-1}+a_{n-1}\cdot b^{n-2}+a_{n-2}\cdot b^{n-3}+\ldots +a_{2}\cdot b^{1}+a_{1}\cdot b^{0}}$with and ${\displaystyle 0\leq a_{i}\leq b-1}$ ${\displaystyle 1\leq i\leq b}$

is a narcissistic number with base b if:

${\displaystyle a_{n}^{n}+a_{n-1}^{n}+a_{n-2}^{n}+\cdots +a_{1}^{n}=a}$.

Examples

Example 1:

The decimal is a narcissistic number with a base . ${\displaystyle 62}$ ${\displaystyle b=4}$

It is in four system (it is ), and actually applies to the then three-digit number: . ${\displaystyle 62=332_{4}}$ ${\displaystyle {\underline {3}}\cdot 4^{2}+{\underline {3}}\cdot 4^{1}+{\underline {2}}\cdot 4^{0}=62}$ ${\displaystyle 62=332_{4}=3^{3}+3^{3}+2^{3}}$

Example 2:

The decimal is a narcissistic number with a base . ${\displaystyle 2292}$ ${\displaystyle b=6}$

It is in the six-system (it is ), and actually applies to the then five-digit number: . ${\displaystyle 2292=14340_{6}}$ ${\displaystyle {\underline {1}}\cdot 6^{4}+{\underline {4}}\cdot 6^{3}+{\underline {3}}\cdot 6^{2}+{\underline {4}}\cdot 6^{1}+{\underline {0}}\cdot 6^{0}=2292}$ ${\displaystyle 2292=14340_{6}=1^{5}+4^{5}+3^{5}+4^{5}+0^{5}}$

A list of the narcissistic numbers with a base has already been given above (sequence A005188 in OEIS ). ${\displaystyle b=10}$

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: ${\displaystyle b}$ ${\displaystyle A=10,B=11,\ldots }$

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. ${\displaystyle 27}$ ${\displaystyle 27}$ ${\displaystyle 53}$ ${\displaystyle 27}$ ${\displaystyle 2}$ ${\displaystyle 53}$ ${\displaystyle 2^{2}+7^{2}=53}$ ${\displaystyle 27}$

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. ${\displaystyle b}$ ${\displaystyle k\cdot (b-1)^{k}}$ ${\displaystyle k\cdot (b-1)^{k}<b^{k-1}}$ ${\displaystyle b}$

• Special case: Every narcissistic number in the decimal system must be less than . ${\displaystyle 10^{60}}$

Proof:

Because of the above property must for k are -digit numbers: . This inequality has the solution . ${\displaystyle k\cdot 9^{k}<10^{k-1}}$ ${\displaystyle k\leq 60}$

Thus, a narcissistic number in the decimal system cannot be greater than . ${\displaystyle 10^{60}}$

• 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:

${\displaystyle 115.132.219.018.763.992.565.095.597.973.971.522.401=1^{39}+1^{39}+5^{39}+\ldots +4^{39}+0^{39}+1^{39}}$

• 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. ${\displaystyle b}$ ${\displaystyle b^{2}+1}$

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). ${\displaystyle b}$ ${\displaystyle \tau (b^{2}+1)-2}$ ${\displaystyle \tau (b)}$ ${\displaystyle b}$ ${\displaystyle \tau (10)=4}$

• 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: ${\displaystyle b\geq 3}$ ${\displaystyle 9}$

2, 72, 90, 108, 153, 270, 423, 450, 531, 558, 630, 648, 738, 1044, 1098, 1125, 1224, 1242, 1287, 1440, 1503, 1566, 1611, 1620, 1800, 1935, ... (Follow A248970 in OEIS )

So there is no three-digit number with these bases . ${\displaystyle xyz_{b}}$ ${\displaystyle x^{3}+y^{3}+z^{3}=x\cdot b^{2}+y\cdot b+z}$

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 . ${\displaystyle b}$ ${\displaystyle b}$

Examples:

• The decimal number has four decimal places, but it can be represented as the sum of five powers of its decimal places: ${\displaystyle 4150}$

${\displaystyle 4150=4^{5}+1^{5}+5^{5}+0^{5}}$

So it is a perfect digital invariant, but not a narcissistic number.

• The smallest PDIs to some power of their digits are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929 , 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153,… (sequence A023052 in OEIS )

The associated potencies are

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7 , 8, 8, 8, 9, 9, 9, 9, ... (Follow A046074 in OEIS )

• In the two upper lists (for example) the numbers 14459929 and 7 are in the 29th position. This means that the 8-digit number

${\displaystyle 14459929=1^{7}+4^{7}+4^{7}+5^{7}+9^{7}+9^{7}+2^{7}+9^{7}}$ 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. ${\displaystyle 1741725=1^{7}+7^{7}+4^{7}+1^{7}+7^{7}+2^{7}+5^{7}}$
• 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):

2, 0, 153, 1634, 4150, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 564240140138, 28116440335967, 0, 4338281769391370, 233411150132317,… (sequence A003321 in OEIS )

For example, it's in the sixth position . That means that is and that applies: ${\displaystyle 548834}$ ${\displaystyle n=6}$ ${\displaystyle 548834=5^{6}+4^{6}+8^{6}+8^{6}+3^{6}+4^{6}}$

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 = . ${\displaystyle a^{1}+b^{2}+c^{3}}$

Examples:

• ${\displaystyle 2427=2^{1}+4^{2}+2^{3}+7^{4}=2+16+8+2401}$
• ${\displaystyle 2646798=2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}}$
• The following numbers are narcissistic in this sense:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539 (series A032799 in OEIS )

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:

${\displaystyle 4624=4^{4}+4^{6}+4^{2}+4^{4}=256+4096+16+256}$

Wild narcissistic numbers

Wild narcissistic numbers are numbers where the way in which they generate themselves from their digits is not uniform.

Example:

${\displaystyle 24739=2^{4}+7!+3^{9}=16+5040+19683}$

Interesting numbers

Interesting numbers are even more free than the wild narcissistic numbers in their generation:

Examples:

${\displaystyle 3456=3!\cdot {\frac {4}{5}}\cdot 6!}$

${\displaystyle 355=3\cdot 5!-5}$

${\displaystyle 127=-1+2^{7}}$

${\displaystyle 343=(3+4)^{3}}$

See also

• Happy number
• Lucky number
• Fortunate number
• Munchausen number

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

1. ^ Armstrong Numbers , Dik T. Winter
2. ^ Lionel Deimel's Web Log
3. ↑ PPDI (Armstrong) Numbers ( Memento of October 27, 2009 in the Internet Archive ), Harvey Heinz
4. ↑ Thomas Jüstel: Special numbers. (PDF) Münster University of Applied Sciences , accessed on October 29, 2014 . 
5. ↑ Eric W. Weisstein : Narcissistic number . In: MathWorld (English).