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A '''self number''', '''Colombian number''' or '''Devlali number''' is an [[integer]] which, in a given [[Radix|base]], cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the [[India]]n [[mathematician]] [[D. R. Kaprekar]].
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The first few base 10 self numbers are:
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: [[1 (number)|1]], [[3 (number)|3]], [[5 (number)|5]], [[7 (number)|7]], [[9 (number)|9]], [[20 (number)|20]], [[31 (number)|31]], [[42 (number)|42]], [[53 (number)|53]], [[64 (number)|64]], [[75 (number)|75]], [[86 (number)|86]], [[97 (number)|97]], [[108 (number)|108]], [[110 (number)|110]], [[121 (number)|121]], [[132 (number)|132]], [[143 (number)|143]], [[154 (number)|154]], 165, [[176 (number)|176]], [[187 (number)|187]], 198, 209, [[211 (number)|211]], [[222 (number)|222]], [[233 (number)|233]], 244, [[255 (number)|255]], 266, [[277 (number)|277]], 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, [[400 (number)|400]], 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525 {{OEIS|id=A003052}}
  <li>[http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=847596 http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=847596]</li>
 
 
A search for self numbers can turn up [[self-descriptive number]]s, which are similar to self numbers in being base-dependent, but quite different in definition and much fewer in frequency.
  <li>[http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/ http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/]</li>
 
 
==Properties==
  <li>[http://sailer.im/forum.php?mod=viewthread&tid=634596&fromuid=31787 http://sailer.im/forum.php?mod=viewthread&tid=634596&fromuid=31787]</li>
In general, for even bases, all [[even and odd numbers|odd]] numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers.<ref name=CS384>Sándor & Crstici (2004) p.384</ref>
 
 
  <li>[http://www.jianyeqiumi.com/forum.php?mod=viewthread&tid=221029&fromuid=60511 http://www.jianyeqiumi.com/forum.php?mod=viewthread&tid=221029&fromuid=60511]</li>
The set of self numbers in a given  base ''q'' is infinite and has a positive [[asymptotic density]]: when ''q'' is odd, this density is 1/2.<ref name=CS385>Sándor & Crstici (2004) p.385</ref>
 
 
  <li>[http://tbekj.cn/home.php?mod=space&uid=4541 http://tbekj.cn/home.php?mod=space&uid=4541]</li>
== Recurrent formula ==
 
 
</ul>
The following [[recurrence relation]] generates some [[Decimal|base 10]] self numbers:
 
:<math>C_k = 8 \cdot 10^{k - 1} + C_{k - 1} + 8</math>
 
(with ''C''<sub>1</sub> = 9)
 
And for [[binary numeral system|binary]] numbers:
 
:<math>C_k = 2^j + C_{k - 1} + 1\,</math>
 
(where ''j'' stands for the number of digits) we can generalize a recurrence relation to generate self numbers in any base ''b'':
 
:<math>C_k = (b - 2)b^{k - 1} + C_{k - 1} + (b - 2)\,</math>
 
in which ''C''<sub>1</sub>&nbsp;=&nbsp;''b''&nbsp;&minus;&nbsp;1 for even bases and ''C''<sub>1</sub>&nbsp;=&nbsp;''b''&nbsp;&minus;&nbsp;2 for odd bases.
 
The existence of these recurrence relations shows that for any base there are infinitely many self numbers.
 
== Self primes ==
 
A '''self prime''' is a self number that is [[prime number|prime]]. The first few self primes are
 
:3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, ... {{OEIS|id=A006378}}
 
In October 2006 [[Luke Pebody]] demonstrated that the largest known [[Mersenne prime]] that is at the same time a self number is 2<sup>24036583</sup>&minus;1. This is then the largest known self prime {{As of|2006|lc=on}}.
 
== Selfness tests ==
=== Reduction tests ===
 
Luke Pebody showed (Oct 2006) that a link can be made between the self property of a large number ''n'' and a low-order portion of that number, adjusted for digit sums:
 
a) In general, ''n'' is self [[if and only if]] ''m'' = R(''n'')+SOD(R(''n''))-SOD(''n'') is self
 
Where:
 
R(''n'') is the smallest rightmost digits of ''n'', greater than 9.d(''n'')
 
d(''n'') is the number of digits in ''n''
 
SOD(''x'') is the sum of digits of ''x'', the function ''S''<sub>10</sub>(''x'') from above.
 
b) If ''n'' = ''a''.10^''b''+''c'', ''c''<10^''b'', then ''n'' is self if and only if both {''m1'' & ''m2''} are negative or self
 
Where:
 
''m1'' = ''c'' - SOD(''a'')
 
''m2'' = SOD(''a''-1)+9.''b''-(''c''+1)
 
c) For the simple case of ''a''=1 & ''c''=0 in the previous model (i.e. ''n''=10^''b''), then ''n'' is self if and only if (9.''b''-1) is self
 
=== Effective test ===
 
Kaprekar [[Demonstration_(proof)|demonstrated]] that:
 
<math>
n \mbox{ is self if }
[ n - DR*(n) - 9 \cdot i ] + SOD([ n - DR*(n) - 9 \cdot i ] ) \neq n
\quad \forall i \in 0 \ldots d(n)
</math>
 
Where:
 
<math>DR*(n) =
\begin{cases}
\frac{DR(n)}{2}, & \mbox{if } DR(n) \mbox{ is even}\\
\frac{DR(n) + 9}{2}, & \mbox{if } DR(n) \mbox{ is odd}
\end{cases}
</math>
 
<math> \begin{align}
DR(n) &{}=
\begin{cases}
9, & \mbox{if } SOD(n) \mod 9 = 0\\
SOD(n) \mod 9, & \mbox{ otherwise}
\end{cases} \\
&{}= (n - 1) \mod 9 + 1
\end{align}</math>
 
<math>SOD(n) \mbox{ is the sum of all digits in } n</math>
 
<math>d(n) \mbox{ is the number of digits in } n</math>
 
== Excerpt from the table of bases where 2007 is self or Colombian ==
 
The following table was calculated in 2007.
{| class="wikitable"
|-
! Base !! Certificate !! Sum of digits
|-
| 40 || <math>1959 = [1, 8, 39]_{40}</math> || 48
|-
| 41 || - || -
|-
| 42 || <math>1967 = [1, 4, 35]_{42}</math> || 40
|-
| 43 || - || -
|-
| 44 || <math>1971 = [1, 0, 35]_{44}</math> || 36
|-
| 44 || <math>1928 = [43, 36]_{44}</math> || 79
|-
| 45 || - || -
|-
| 46 || <math>1926 = [41, 40]_{46}</math> || 81
|-
| 47 || - || -
|-
| 48 || - || -
|-
| 49 || - || -
|-
| 50 || <math>1959 = [39, 9]_{50}</math> || 48
|-
| 51 || - || -
|-
| 52 || <math>1947 = [37, 23]_{52}</math> || 60
|-
| 53 || - || -
|-
| 54 || <math>1931 = [35, 41]_{54}</math> || 76
|-
| 55 || - || -
|-
| 56 || <math>1966 = [35, 6]_{56}</math> || 41
|-
| 57 || - || -
|-
| 58 || <math>1944 = [33, 30]_{58}</math> || 63
|-
| 59 || - || -
|-
| 60 || <math>1918 = [31, 58]_{60}</math> || 89
|}
 
==References==
{{reflist}}
* Kaprekar, D. R. ''The Mathematics of New Self-Numbers'' Devaiali  (1963): 19 - 20.
* {{cite journal |author=R. B. Patel |title=Some Tests for ''k''-Self Numbers |journal=Math. Student |volume=56 |year=1991 |pages=206–210}}
* {{cite journal |author=B. Recaman |title=Problem E2408 |journal=Amer. Math. Monthly |volume=81 |issue=4 |year=1974 |pages=407 |doi=10.2307/2319017}}
* {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=32–36 | zbl=1079.11001 }}
* {{MathWorld|urlname=SelfNumber|title=Self Number}}
 
{{Prime number classes}}
{{Classes of natural numbers}}
 
[[Category:Base-dependent integer sequences]]

Revision as of 04:07, 26 February 2014

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