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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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Irwin Butts is what my spouse enjoys to call me though I don't truly like becoming known as like that. Puerto Rico is exactly where he and his wife reside. For years I've been operating as a payroll clerk. Body developing is 1 of the issues I love most.
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