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| | | Hi there, I am Alyson Pomerleau and I believe it seems fairly good when you say it. Mississippi is where his house is. Office supervising is where her main income comes from but she's already applied for another 1. To climb is something she would never give up.<br><br>Here is my weblog; real psychic readings ([http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 galab-work.cs.pusan.ac.kr]) |
| In [[Recreational mathematics|recreational]] [[number theory]], a '''narcissistic number'''<ref name="mw">{{MathWorld |title=Narcissistic Number |urlname=NarcissisticNumber}}</ref><ref name="moore">[http://www.cs.umd.edu/Honors/reports/NarcissisticNums/NarcissisticNums.html ''Perfect and PluPerfect Digital Invariants''] by Scott Moore</ref> (also known as a '''pluperfect digital invariant''' ('''PPDI'''),<ref>[http://web.archive.org/web/20091027123639/http://www.geocities.com/~harveyh/narciss.htm PPDI (Armstrong) Numbers] by Harvey Heinz</ref> an '''Armstrong number'''<ref>[http://homepages.cwi.nl/~dik/english/mathematics/armstrong.htm Armstrong Numbersl] by Dik T. Winter</ref> (after Michael F. Armstrong)<ref>[http://blog.deimel.org/2010/05/mystery-solved.html Lionel Deimel’s Web Log]</ref> or a '''plus perfect number''')<ref>{{OEIS|id=A005188}}</ref> is a number that is the sum of its own digits each raised to the power of the number of digits. This definition depends on the base ''b'' of the number system used, e.g. ''b'' = 10 for the [[decimal|decimal system]] or ''b'' = 2 for the [[binary numeral system|binary system]].
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| ==Definition==
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| The definition of a narcissistic number relies on the decimal representation ''n'' = ''d''<sub>''k''</sub>''d''<sub>''k''-1</sub>...''d''<sub>1</sub> of a [[natural number]] ''n'', i.e.
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| :''n'' = ''d''<sub>''k''</sub>·10<sup>''k''-1</sup> + ''d''<sub>''k''-1</sub>·10<sup>''k''-2</sup> + ... + ''d''<sub>2</sub>·10 + ''d''<sub>1</sub>,
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| with ''k'' digits ''d''<sub>''i''</sub> satisfying 0 ≤ ''d''<sub>''i''</sub> ≤ 9. Such a number ''n'' is called narcissistic if it satisfies the condition
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| :''n'' = ''d''<sub>''k''</sub><sup>''k''</sup> + ''d''<sub>''k''-1</sub><sup>''k''</sup> + ... + ''d''<sub>2</sub><sup>''k''</sup> + ''d''<sub>1</sub><sup>''k''</sup>.
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| For example the 3-digit decimal number 153 is a narcissistic number because 153 = 1<sup>3</sup> + 5<sup>3</sup> + 3<sup>3</sup>.
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| Narcissistic numbers can also be defined with respect to [[numeral system]]s with a base ''b'' other than ''b'' = 10. The base-''b'' representation of a natural number ''n'' is defined by
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| :''n'' = ''d''<sub>''k''</sub>''b''<sup>''k''-1</sup> + ''d''<sub>''k''-1</sub>''b''<sup>''k''-2</sup> + ... + ''d''<sub>2</sub>''b'' + ''d''<sub>1</sub>,
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| where the base-''b'' digits ''d''<sub>''i''</sub> satisfy the condition 0 ≤ ''d''<sub>i</sub> ≤ ''b''-1. | |
| For example the (decimal) number 17 is a narcissistic number with respect to the numeral system with base ''b'' = 3. Its three base-3 digits are 122, because 17 = 1·3<sup>2</sup> + 2·3 + 2 , and it satisfies the equation 17 = 1<sup>3</sup> + 2<sup>3</sup> + 2<sup>3</sup>.
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| If the constraint that the power must equal the number of digits is dropped, so that for some ''m'' possibly different from ''k'' it happens that
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| :''n'' = ''d''<sub>''k''</sub><sup>''m''</sup> + ''d''<sub>''k''-1</sub><sup>''m''</sup> + ... + ''d''<sub>2</sub><sup>''m''</sup> + ''d''<sub>1</sub><sup>''m''</sup>,
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| then ''n'' is called a '''perfect digital invariant''' or '''PDI'''.<ref>[http://web.archive.org/web/20091027123639/http://www.geocities.com/~harveyh/narciss.htm PDIs] by Harvey Heinz</ref><ref name="moore"/> For example, the decimal number 4150 has four decimal digits and is the sum of the ''fifth'' powers of its decimal digits
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| :4150 = 4<sup>5</sup> + 1<sup>5</sup> + 5<sup>5</sup> + 0<sup>5</sup>,
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| so it is a perfect digital invariant but ''not'' a narcissistic number.
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| In "[[A Mathematician's Apology]]", [[G. H. Hardy]] wrote:
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| :''There are just four numbers, after unity, which are the sums of the cubes of their digits:''
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| ::<math>153=1^3+5^3+3^3</math>
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| ::<math>370=3^3+7^3+0^3</math>
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| ::<math> 371=3^3+7^3+1^3</math>
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| ::<math>407=4^3+0^3+7^3</math>.<br>
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| :''These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.''
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| ==Narcissistic numbers in various bases==
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| The sequence of "base 10" narcissistic numbers starts:
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| [[0 (number)|0]], [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[5 (number)|5]], [[6 (number)|6]], [[7 (number)|7]], [[8 (number)|8]], [[9 (number)|9]], [[153 (number)|153]], [[370 (number)|370]], [[371 (number)|371]], 407, 1634, 8208, 9474 ... {{OEIS|id=A005188}}
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| The sequence of "base 3" narcissistic numbers starts:
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| 0, 1, 2, [[5 (number)|12]], [[8 (number)|22]], [[17 (number)|122]]
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| The sequence of "base 4" narcissistic numbers starts:
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| 0, 1, 2, 3, [[28 (number)|130]], [[29 (number)|131]], [[35 (number)|203]], [[43 (number)|223]], [[55 (number)|313]]
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| The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the ''k''th powers of a ''k'' digit number in base ''b'' is
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| :<math>k(b-1)^k\, ,</math>
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| and if ''k'' is large enough then
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| :<math>k(b-1)^k<b^{k-1}\, ,</math>
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| in which case no base ''b'' narcissistic number can have ''k'' or more digits. Setting ''b'' equal to 10 shows that the largest narcissistic number in base 10 must be less than 10<sup>60</sup>.<ref name="mw" />
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| There are only 88 narcissistic numbers in base 10, of which the largest is
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| :115,132,219,018,763,992,565,095,597,973,971,522,401
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| with 39 digits.<ref name="mw" />
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| Unlike narcissistic numbers, no upper bound can be determined for the size of PDIs in a given base, and it is not currently known whether or not the number of PDIs for an arbitrary base is finite or infinite.<ref name="moore"/>
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| ==Related concepts==
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| The term "narcissistic number" is sometimes used in a wider sense to mean a number that is equal to any mathematical manipulation of its own digits. With this wider definition narcisstic numbers include:
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| * [[Constant base numbers]] : <math>n=m^{d_k} + m^{d_{k-1}} + \dots + m^{d_2} + m^{d_1}</math> for some ''m''.
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| * [[Perfect digit-to-digit invariants]] {{OEIS|id=A046253}} : <math>n = d_k^{d_k} + d_{k-1}^{d_{k-1}} + \dots + d_2^{d_2} + d_1^{d_1}\, ,\text{ e.g. } 3435 = 3^3 + 4^4 + 3^3 + 5^5\, .</math>
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| * [[Ascending power numbers]] {{OEIS|id=A032799}} : <math>n = d_k^1 + d_{k-1}^2 + \dots + d_2^{k-1} + d_1^k\, ,\text{ e.g. } 135 = 1^1 + 3^2 + 5^3 \, .</math>
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| * [[Friedman number]]s {{OEIS|id=A036057}}.
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| * Radical narcissistic numbers {{OEIS|id= A119710}} <ref>Rose, Colin (2005), Radical Narcissistic Numbers, Journal of Recreational Mathematics, 33(4), pages 250-254.</ref> <math>{ e.g. } 729 = (7+2)^{\sqrt{9}}, \text{ }4096=\sqrt{\sqrt{\sqrt{\sqrt{4}+0}}}^{96}</math>
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| * Pretty wild narcissistic numbers <ref>[http://www.numq.com/pwn/ Pretty wild narcissistic numbers - numbers that pwn] by Colin Rose</ref>
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| * [[Sum-product number]]s {{OEIS|id=A038369}} : <math>n=\left(\sum_{i=1}^{k}{d_i}\right) \left(\prod_{i=1}^{k}{d_i}\right) \, ,\text{ e.g. } 144 = (1+4+4) \times (1 \times4 \times 4) \, .</math>
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| * [[Dudeney number]]s {{OEIS|id=A061209}} :<math>n=\left(\sum_{i=1}^{k}{d_i}\right)^3\, ,\text{ e.g. } 512 = (5+1+2)^3 \, .</math>
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| * [[Factorion]]s {{OEIS|id=A014080}} :<math>n=\sum_{i=1}^{k}{d_i}!\, ,\text{ e.g. } 145 = 1! + 4! + 5! \, .</math>
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| where ''d''<sub>''i''</sub> are the digits of ''n'' in some base.
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * [[Joseph Madachy|Joseph S. Madachy]], ''Mathematics on Vacation'', Thomas Nelson & Sons Ltd. 1966, pages 163-175.
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| * Rose, Colin (2005), ''Radical narcissistic numbers'', Journal of Recreational Mathematics, 33(4), 2004-2005, pages 250-254.
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| * [http://www.mathews-archive.com/digit-related-numbers/pdi.html ''Perfect Digital Invariants''] by Walter Schneider
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| * [http://arxiv.org/abs/0911.3038 ''On a curious property of 3435''] by Daan van Berkel
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| {{refend}}
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| ==External links==
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| * [http://www.deimel.org/rec_math/DI_0.htm Digital Invariants]
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| * [http://everything2.net/index.pl?node_id=1407017&displaytype=printable&lastnode_id=1407017 Armstrong Numbers]
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| * [http://www.cs.mtu.edu/~shene/COURSES/cs201/NOTES/chap04/arms.html Armstrong numbers between 1-999 calculator]
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| * {{cite web|last=Symonds|first=Ria|title=153 ♥ Narcissistic Number|url=http://www.numberphile.com/videos/153_narcissistic.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| {{Classes of natural numbers}}
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| [[Category:Base-dependent integer sequences]]
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| [[Category:Recreational mathematics]]
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