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| An ''n''-'''parasitic number''' (in base 10) is a positive [[natural number]] which can be [[multiplication|multiplied]] by ''n'' by moving the rightmost [[Numerical digit|digit]] of its [[decimal|decimal representation]] to the front. Here ''n'' is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right [[circular shift]] by one place. For example, 4•128205=512820, so 128205 is 4-parasitic. Most authors do not allow leading zeros to be used, and this article follows that convention. So even though 4•025641=102564, the number 025641 is ''not'' 4-parasitic.
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| == Derivation ==
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| An ''n''-parasitic number can be derived by starting with a digit ''k'' (which should be equal to ''n'' or greater) in the rightmost (units) place, and working up one digit at a time.
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| For example, for ''n'' = 4 and ''k'' = 7:
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| :4•7=2'''8'''
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| :4•87=3'''48'''
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| :4•487=1'''948'''
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| :4•9487=3'''7948'''
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| :4•79487=3'''17948'''
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| :4•179487='''717948'''.
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| So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487 etc.
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| Notice that the [[repeating decimal]]
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| :<math>x=0.179487179487179487\ldots=0.\overline{179487} \mbox{ has }4x=0.\overline{717948}=\frac{7.\overline{179487}}{10}.</math>
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| Thus
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| :<math>4x=\frac{7+x}{10} \mbox{ so } x=\frac{7}{39}.</math>
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| In general, an ''n''-parasitic number can be found as follows. Pick a one digit integer ''k'' such that {{nowrap|''k'' ≥ ''n''}}, and take the period of the [[repeating decimal]] ''k''/(10''n''−1).
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| This will be <math> \frac{k}{10n-1}(10^m-1)</math>
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| where ''m'' is the length of the period; i.e. the [[multiplicative order]] of 10 [[modular arithmetic|modulo]] {{nowrap|(10''n'' − 1)}}.
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| For another example, if ''n'' = 2, then 10''n'' − 1 = 19 and the repeating decimal for 1/19 is
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| : <math>\frac{1}{19}=0.\overline{052631578947368421}.</math>
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| So that for 2/19 is double that:
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| : <math>\frac{2}{19}=0.\overline{105263157894736842}.</math>
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| The length ''m'' of this period is 18, the same as the order of 10 modulo 19, so {{nowrap|2 × (10<sup>18</sup> − 1)/19}} = 105263157894736842.
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| 105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.
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| == Smallest n-parasitic numbers ==
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| The smallest ''n''-parasitic numbers are also known as '''Dyson numbers''', after a puzzle concerning these numbers posed by [[Freeman Dyson]].<ref>{{citation|title=The Civil Heretic|journal=[[New York Times Magazine]]|url=http://www.nytimes.com/2009/03/29/magazine/29Dyson-t.html|first=Nicholas|last=Dawidoff|date=March 25, 2009}}.</ref><ref>{{citation|title=Freeman Dyson’s 4th-Grade Math Puzzle|journal=[[New York Times]]|first=John|last=Tierney|date=April 6, 2009|url=http://tierneylab.blogs.nytimes.com/2009/04/06/freeman-dysons-4th-grade-math-puzzle/}}.</ref><ref>{{citation|title=Prize for Dyson Puzzle|journal=[[New York Times]]|first=John|last=Tierney|date=April 13, 2009|url=http://tierneylab.blogs.nytimes.com/2009/04/13/prize-for-dyson-puzzle/}}.</ref> They are:
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| <table border="1" cellpadding="2">
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| <tr><td width=30>''n''<td>Smallest ''n''-parasitic number <td> period of
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| <tr><td>1<td>1<td>1/9
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| <tr><td>2<td>105263157894736842<td>2/19
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| <tr><td>3<td>1034482758620689655172413793<td>3/29
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| <tr><td>4<td>102564<td>4/39
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| <tr><td>5<td>[[142857]]<td>'''7'''/49=1/7
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| <tr><td>6<td>1016949152542372881355932203389830508474576271186440677966<td>6/59
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| <tr><td>7<td>1014492753623188405797<td>7/69
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| <tr><td>8<td>1012658227848<td>8/79
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| <tr><td>9<td>10112359550561797752808988764044943820224719<td>9/89
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| </table>
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| ==General note==
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| In general, if we relax the rules to allow a leading zero, then there are 9 ''n''-parasitic numbers for each ''n''. Otherwise only if ''k'' ≥ ''n'' then the numbers do not start with zero and hence fit the actual definition.
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| Other ''n''-parasitic integers can be built by concatenation. For example, since 179487 is a 4-parasitic number, so are 179487179487, 179487179487179487 etc.
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| ==See also==
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| *[[Cyclic number]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *[[Clifford A. Pickover|C. A. Pickover]], ''Wonders of Numbers'', Chapter 28, [[Oxford University Press]] UK, 2000.
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| *Sequence {{OEIS2C|A092697}} in the [[On-Line Encyclopedia of Integer Sequences]].
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| {{Classes of natural numbers}}
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| [[Category:Base-dependent integer sequences]]
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Hello and welcome. My name is Irwin and I totally dig that title. North Dakota is our birth place. Managing people is his occupation. One of the extremely very best issues in the globe for me is to do aerobics and now I'm trying to earn money with it.
Feel free to surf to my blog post; at home std test