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{{Short description|Type of natural number}}
In [[Number theory|number theory]], a '''self number''' or '''Devlali number''' in a given number base <math>b</math> is a [[Natural number|natural number]] that cannot be written as the sum of any other natural number <math>n</math> and the individual digits of <math>n</math>. 20 is a self number (in base 10), because no such combination can be found (all <math>n < 15</math> give a result less than 20; all other <math>n</math> give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 using ''n'' = 15. These numbers were first described in 1949 by the {{wipe|India}}n [[Biography:Mathematician|mathematician]] [[Biography:D. R. Kaprekar|D. R. Kaprekar]].

==Definition and properties==
Let <math>n</math> be a natural number. We define the <math>b</math>-'''self function''' for base <math>b > 1</math> <math>F_b : \mathbb{N} \rightarrow \mathbb{N}</math> to be the following:
:<math>F_{b}(n) = n + \sum_{i=0}^{k - 1} d_i. </math>
where <math>k = \lfloor \log_{b}{n} \rfloor + 1</math> is the number of digits in the number in base <math>b</math>, and 
:<math>d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i}</math>
is the value of each digit of the number. A natural number <math>n</math> is a <math>b</math>-'''self number''' if the [[Inverse function#Preimages|preimage]] of <math>n</math> for <math>F_b</math> is the [[Empty set|empty set]].

In general, for even bases, all 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>

The set of self numbers in a given  base <math>b</math> is infinite and has a positive asymptotic density: when <math>b</math> is odd, this density is 1/2.<ref name=CS385>Sándor & Crstici (2004) p.385</ref>

== Recurrent formula ==
The following [[Recurrence relation|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 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.

== 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:
{{ordered list|list_style_type=lower-alpha
|1=In general, ''n'' is self [[If and only if|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.

|2=If {{tmath|1=n = a\cdot 10^b+c,\ c<10^b}}, then ''n'' is self if and only if both {''m''{{sub|1}} & ''m''{{sub|2}}} are negative or self

Where:

:''m''{{sub|1}} = ''c'' - SOD(''a'')

:''m''{{sub|2}} = SOD(''a''-1)+9&middot;''b''-(''c''+1)

|3=For the simple case of ''a''=1 & ''c''=0 in the previous model (i.e. {{tmath|1=n=10^b}}), then ''n'' is self if and only if (9&middot;''b''-1) is self
}}
=== Effective test ===

Kaprekar demonstrated that:

:{{mvar|n}} is self if <math> \mathrm{SOD}(| n - \mathrm{DR}^*(n) - 9 \cdot i | ) \neq [\mathrm{DR}^*(n) + 9 \cdot i ]
\quad \forall i \in 0 \ldots d(n)
</math>

Where:

:<math>\mathrm{DR}^*(n) =
\begin{cases}
 \frac{\mathrm{DR}(n)}{2}, & \text{if } \mathrm{DR}(n) \text{ is even}\\
 \frac{\mathrm{DR}(n) + 9}{2}, & \text{if } \mathrm{DR}(n) \text{ is odd}
\end{cases}
</math>

:<math> \begin{align}
\mathrm{DR}(n) &{}=
\begin{cases}
 9, & \text{if } \mathrm{SOD}(n) \mod 9 = 0\\
 \mathrm{SOD}(n) \mod 9, & \text{otherwise}
\end{cases} \\
&{}= 1+[(n - 1) \mod 9]
\end{align}</math>

:<math>\mathrm{SOD}(n)</math> is the sum of all digits in {{mvar|n}}.
:<math>d(n)</math> is the number of digits in {{mvar|n}}.

== Self numbers in specific bases ==
For [[Binary number|base 2]] self numbers, see {{oeis|id=A010061}}. (written in base 10)

The first few base 10 self numbers are:
: 1, 3, 5, 7, 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 (number)|165]], [[176 (number)|176]], [[187 (number)|187]], 198, [[209 (number)|209]], [[211 (number)|211]], [[222 (number)|222]], [[233 (number)|233]], [[244 (number)|244]], [[255 (number)|255]], [[266 (number)|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, ... {{OEIS|id=A003052}}

In [[Duodecimal|base 12]], the self numbers are: (using A and B for ten and eleven, respectively)
:1, 3, 5, 7, 9, B, 20, 31, 42, 53, 64, 75, 86, 97, A8, B9, 102, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1A9, 1BA, 20B, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2AA, 2BB, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39A, 3AB, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48A, 49B, 4B0, 501, 512, 514, 525, 536, 547, 558, 569, 57A, 58B, 5A0, 5B1, ...

== Self primes ==

A '''self prime''' is a self number that is [[Prime number|prime]]. 

The first few self primes in base 10 are
:3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, ... {{OEIS|id=A006378}}

The first few self primes in base 12 are:
:3, 5, 7, B, 31, 75, 255, 277, 2AA, 3BA, 435, 457, 58B, 5B1, ...

In October 2006 Luke Pebody demonstrated that the largest known [[Mersenne prime]] in base 10 that is at the same time a self number is 2<sup>24036583</sup>&minus;1. This is then the largest known self prime in base 10 {{As of|2006|lc=on}}.

==Extension to negative integers==
Self numbers can be extended to the negative integers by use of a [[Signed-digit representation|signed-digit representation]] to represent each integer.

== Excerpt from the table of bases where 2007 is self ==

The following table was calculated in 2007.
{| class="wikitable"
|-
! Base !! Certificate !! Sum of digits
|-
| style="text-align:right" | 40 || <math>1959 = [1, 8, 39]_{40}</math> || style="text-align:right" | 48
|-
| style="text-align:right" | 41 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 42 || <math>1967 = [1, 4, 35]_{42}</math> || style="text-align:right" | 40
|-
| style="text-align:right" | 43 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 44 || <math>1971 = [1, 0, 35]_{44}</math> || style="text-align:right" | 36
|-
| style="text-align:right" | 44 || <math>1928 = [43, 36]_{44}</math> || style="text-align:right" | 79
|-
| style="text-align:right" | 45 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 46 || <math>1926 = [41, 40]_{46}</math> || style="text-align:right" | 81
|-
| style="text-align:right" | 47 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 48 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 49 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 50 || <math>1959 = [39, 9]_{50}</math> || style="text-align:right" | 48
|-
| style="text-align:right" | 51 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 52 || <math>1947 = [37, 23]_{52}</math> || style="text-align:right" | 60
|-
| style="text-align:right" | 53 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 54 || <math>1931 = [35, 41]_{54}</math> || style="text-align:right" | 76
|-
| style="text-align:right" | 55 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 56 || <math>1966 = [35, 6]_{56}</math> || style="text-align:right" | 41
|-
| style="text-align:right" | 57 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 58 || <math>1944 = [33, 30]_{58}</math> || style="text-align:right" | 63
|-
| style="text-align:right" | 59 || {{sdash}} || {{sdash}}
|-
| style="text-align:right" | 60 || <math>1918 = [31, 58]_{60}</math> || style="text-align:right" | 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|jstor=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]]
[[Category:Inverse functions]]

{{Sourceattribution|Self number}}