{{위키데이터 속성 추적}}
'''수단 함수'''(Sudan function)는 [[계산 이론]]에서 [[재귀]]적이지만 [[원시 재귀 함수|원시 재귀]]적이지는 않은 [[함수]]의 예이다. 이는 더 잘 알려진 [[아커만 함수]]의 경우에도 마찬가지이다.

1926년에 [[다비트 힐베르트]]는 모든 계산 가능한 함수가 원시 재귀 함수라고 추측했다. 이는 그의 제자인 [[가브리엘 수단]]과 [[빌헬름 아커만]]에 의해 빠르게 연속해서 출판된 다양한 기능을 사용하여 반박되었다. 수단은 1927년, 아커만은 1928년에 발표되었다.

수단 함수는 원시 재귀가 아닌 재귀 함수의 가장 먼저 발표된 예이다.

== 정의 ==
:<math>\begin{array}{lll}
F_0 (x, y) & = x+y \\
F_{n+1} (x, 0) & = x & \text{if } n \ge 0 \\
F_{n+1} (x, y+1) & = F_n (F_{n+1} (x, y), F_{n+1} (x, y) + y + 1) & \text{if } n\ge 0 \\
\end{array}</math>
마지막 방정식은 다음과 같이 동등하게 쓸 수 있다.
:<math>\begin{array}{lll}
F_{n+1} (x, y+1) & = F_n(F_{n+1} (x, y), F_0(F_{n+1}(x, y), y + 1) \\
\end{array}</math>.

== 값 표 ==

=== F<sub>0</sub>의 값 ===
''F''<sub>0</sub>(''x'',&nbsp;''y'') = x&nbsp;+&nbsp;y
{| class="wikitable" style="text-align:right; font-size:100%"
|-
! <big>''<sub>y</sub>''&nbsp;\&nbsp;''<sup>x</sup>''</big> !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
|-
! 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10
|-
! 1
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11
|-
! 2
| 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12
|-
! 3
| 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13
|-
! 4
| 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14
|-
! 5
| 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15
|-
! 6
| 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|-
! 7
| 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17
|-
! 8
| 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18
|-
! 9
| 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19
|-
! 10
| 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20
|-
|}

=== F<sub>1</sub>의 값 ===
''F''<sub>1</sub>(''x'',&nbsp;''y'') = 2<sup>y</sup>&nbsp;·&nbsp;(x&nbsp;+&nbsp;2) − y − 2
{| class="wikitable" style="text-align:right; font-size:95%"
|-
! <big>''<sub>y</sub>''&nbsp;\&nbsp;''<sup>x</sup>''</big> !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
|-
! 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10
|-
! 1
| 1 || 3 || 5 || 7 || 9 || 11 || 13 || 15 || 17 || 19 || 21
|-
! 2
| 4 || 8 || 12 || 16 || 20 || 24 || 28 || 32 || 36 || 40 || 44
|-
! 3
| 11 || 19 || 27 || 35 || 43 || 51 || 59 || 67 || 75 || 83 || 91
|-
! 4
| 26 || 42 || 58 || 74 || 90 || 106 || 122 || 138 || 154 || 170 || 186
|-
! 5
| 57 || 89 || 121 || 153 || 185 || 217 || 249 || 281 || 313 || 345 || 377
|-
! 6
| 120 || 184 || 248 || 312 || 376 || 440 || 504 || 568 || 632 || 696 || 760
|-
! 7
| 247 || 375 || 503 || 631 || 759 || 887 || 1015 || 1143 || 1271 || 1399 || 1527
|-
! 8
| 502 || 758 || 1014 || 1270 || 1526 || 1782 || 2038 || 2294 || 2550 || 2806 || 3062
|-
! 9
| 1013 || 1525 || 2037 || 2549 || 3061 || 3573 || 4085 || 4597 || 5109 || 5621 || 6133
|-
! 10
| 2036 || 3060 || 4084 || 5108 || 6132 || 7156 || 8180 || 9204 || 10228 || 11252 || 12276
|-
|}

=== F<sub>2</sub>의 값 ===
{| class="wikitable" style="text-align:right; font-size:90%"
|-
! <big>''<sub>y</sub>''&nbsp;\&nbsp;''<sup>x</sup>''</big> !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7
|-
! rowspan="2" | 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7
|-
| colspan="8" align="center" style="background:#FAF7F4" | ''x''
|-
! rowspan="4" | 1
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(0,&nbsp;0), &nbsp;<br>F<sub>2</sub>(0,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(1,&nbsp;0), &nbsp;<br>F<sub>2</sub>(1,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(2,&nbsp;0), &nbsp;<br>F<sub>2</sub>(2,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(3,&nbsp;0), &nbsp;<br>F<sub>2</sub>(3,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(4,&nbsp;0), &nbsp;<br>F<sub>2</sub>(4,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(5,&nbsp;0), &nbsp;<br>F<sub>2</sub>(5,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(6,&nbsp;0), &nbsp;<br>F<sub>2</sub>(6,&nbsp;0)+1)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(7,&nbsp;0), &nbsp;<br>F<sub>2</sub>(7,&nbsp;0)+1)
|-
| F<sub>1</sub>(0,&nbsp;1)
| F<sub>1</sub>(1,&nbsp;2)
| F<sub>1</sub>(2,&nbsp;3)
| F<sub>1</sub>(3,&nbsp;4)
| F<sub>1</sub>(4,&nbsp;5)
| F<sub>1</sub>(5,&nbsp;6)
| F<sub>1</sub>(6,&nbsp;7)
| F<sub>1</sub>(7,&nbsp;8)
|-
| 1 || 8 || 27 || 74 || 185 || 440 || 1015 || 2294
|-
| colspan="8" align="center" style="background:#FAF7F4" | ''2<sup>x+1</sup>&nbsp;·&nbsp;(x + 2) − x − 3''<br>≈ 10<sup>lg&nbsp;2·(x+1) + lg(x+2)</sup>
|-
! rowspan="4" | 2
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(0,&nbsp;1), &nbsp;<br>F<sub>2</sub>(0,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(1,&nbsp;1), &nbsp;<br>F<sub>2</sub>(1,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(2,&nbsp;1), &nbsp;<br>F<sub>2</sub>(2,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(3,&nbsp;1), &nbsp;<br>F<sub>2</sub>(3,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(4,&nbsp;1), &nbsp;<br>F<sub>2</sub>(4,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(5,&nbsp;1), &nbsp;<br>F<sub>2</sub>(5,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(6,&nbsp;1), &nbsp;<br>F<sub>2</sub>(6,&nbsp;1)+2)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(7,&nbsp;1), &nbsp;<br>F<sub>2</sub>(7,&nbsp;1)+2)
|- 
| F<sub>1</sub>(1,&nbsp;3)
| F<sub>1</sub>(8,&nbsp;10)
| F<sub>1</sub>(27,&nbsp;29)
| F<sub>1</sub>(74,&nbsp;76)
| F<sub>1</sub>(185,&nbsp;187)
| F<sub>1</sub>(440,&nbsp;442)
| F<sub>1</sub>(1015,&nbsp;1017)
| F<sub>1</sub>(2294,&nbsp;2296)
|-
| 19
| 10228
| 15569256417
| ≈ 5,742397643&nbsp;·&nbsp;10<sup>24</sup>
| ≈ 3,668181327&nbsp;·&nbsp;10<sup>58</sup>
| ≈ 5,019729940&nbsp;·&nbsp;10<sup>135</sup>
| ≈ 1,428323374&nbsp;·&nbsp;10<sup>309</sup>
| ≈ 3,356154368&nbsp;·&nbsp;10<sup>694</sup>
|-
| colspan="8" align="center" style="background:#FAF7F4" | 2<sup>2<sup>x+1</sup>·(x+2) − x − 1</sup> · (2<sup>x+1</sup>·(x+2) − x − 1) − (2<sup>x+1</sup>·(x+2) − x + 1)<br>≈ 10<sup>lg&nbsp;2 · (2<sup>x+1</sup>·(x+2) − x − 1) + lg(2<sup>x+1</sup>·(x+2) − x − 1)</sup>  ≈ 10<sup>lg&nbsp;2 · 2<sup>x+1</sup>·(x+2) + lg(2<sup>x+1</sup>·(x+2))</sup>  ≈ 10<sup>lg&nbsp;2 · (2<sup>x+1</sup>·(x+2))</sup> = 10<sup>10<sup>lg&nbsp;lg&nbsp;2 + lg&nbsp;2·(x+1) + lg(x+2)</sup></sup> ≈ 10<sup>10<sup>lg&nbsp;2·(x+1) + lg(x+2)</sup></sup>
|-
! rowspan="5" | 3
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(0,&nbsp;2), &nbsp;<br>F<sub>2</sub>(0,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(1,&nbsp;2), &nbsp;<br>F<sub>2</sub>(1,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(2,&nbsp;2), &nbsp;<br>F<sub>2</sub>(2,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(3,&nbsp;2), &nbsp;<br>F<sub>2</sub>(3,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(4,&nbsp;2), &nbsp;<br>F<sub>2</sub>(4,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(5,&nbsp;2), &nbsp;<br>F<sub>2</sub>(5,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(6,&nbsp;2), &nbsp;<br>F<sub>2</sub>(6,&nbsp;2)+3)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(7,&nbsp;2), &nbsp;<br>F<sub>2</sub>(7,&nbsp;2)+3)
|-
| F<sub>1</sub>(F<sub>1</sub>(1,3), &nbsp;<br>F<sub>1</sub>(1,3)+3)
| F<sub>1</sub>(F<sub>1</sub>(8,10), &nbsp;<br>F<sub>1</sub>(8,10)+3)
| F<sub>1</sub>(F<sub>1</sub>(27,29), &nbsp;<br>F<sub>1</sub>(27,29)+3)
| F<sub>1</sub>(F<sub>1</sub>(74,76), &nbsp;<br>F<sub>1</sub>(74,76)+3)
| F<sub>1</sub>(F<sub>1</sub>(185,187), &nbsp;<br>F<sub>1</sub>(185,187)+3)
| F<sub>1</sub>(F<sub>1</sub>(440,442), &nbsp;<br>F<sub>1</sub>(440,442)+3)
| F<sub>1</sub>(F<sub>1</sub>(1015,1017), &nbsp;<br>F<sub>1</sub>(1015,1017)+3)
| F<sub>1</sub>(F<sub>1</sub>(2294,2297), &nbsp;<br>F<sub>1</sub>(2294,2297)+3)
|-
| F<sub>1</sub>(19,&nbsp;22)
| F<sub>1</sub>(10228,&nbsp;10231)
| F<sub>1</sub>(15569256417,<br>15569256420)
| F<sub>1</sub>(≈6·10<sup>24</sup>,&nbsp;≈6·10<sup>24</sup>)
| F<sub>1</sub>(≈4·10<sup>58</sup>,&nbsp;≈4·10<sup>58</sup>)
| F<sub>1</sub>(≈5·10<sup>135</sup>,&nbsp;≈5·10<sup>135</sup>)
| F<sub>1</sub>(≈10<sup>309</sup>,&nbsp;≈10<sup>309</sup>)
| F<sub>1</sub>(≈3·10<sup>694</sup>,&nbsp;≈3·10<sup>694</sup>)
|-
| 88080360
| ≈ 7,04&nbsp;·&nbsp;10<sup>3083</sup>
| ≈ 7,82&nbsp;·&nbsp;10<sup>4686813201</sup>
| ≈ 10<sup>1,72·10<sup>24</sup></sup>
| ≈ 10<sup>1,10·10<sup>58</sup></sup>
| ≈ 10<sup>1,51·10<sup>135</sup></sup>
| ≈ 10<sup>4,30·10<sup>308</sup></sup>
| ≈ 10<sup>1,01·10<sup>694</sup></sup>
|-
| colspan="8" align="center" style="background:#FAF7F4" | longer expression, starts with 2<sup>2<sup>2<sup>x+1</sup></sup></sup> an, ≈ 10<sup>10<sup>10<sup>lg&nbsp;2·(x+1) + lg(x+2)</sup></sup></sup>
|-
! rowspan="5" | 4
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(0,&nbsp;3), &nbsp;<br>F<sub>2</sub>(0,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(1,&nbsp;3), &nbsp;<br>F<sub>2</sub>(1,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(2,&nbsp;3), &nbsp;<br>F<sub>2</sub>(2,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(3,&nbsp;3), &nbsp;<br>F<sub>2</sub>(3,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(4,&nbsp;3), &nbsp;<br>F<sub>2</sub>(4,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(5,&nbsp;3), &nbsp;<br>F<sub>2</sub>(5,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(6,&nbsp;3), &nbsp;<br>F<sub>2</sub>(6,&nbsp;3)+4)
| F<sub>1</sub>&nbsp;(F<sub>2</sub>(7,&nbsp;3), &nbsp;<br>F<sub>2</sub>(7,&nbsp;3)+4)
|-
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(19,&nbsp;22), &nbsp;<br>F<sub>1</sub>(19,&nbsp;22)+4)
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(10228, &nbsp;<br>10231), &nbsp;<br>F<sub>1</sub>(10228, &nbsp;<br>10231)+4)
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(15569256417, &nbsp;<br>15569256420), &nbsp;<br>F<sub>1</sub>(15569256417, &nbsp;<br>15569256420)+4)
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(≈5,74·10<sup>24</sup>,&nbsp;≈5,74·10<sup>24</sup>),<br>F<sub>1</sub>(≈5,74·10<sup>24</sup>,&nbsp;≈5,74·10<sup>24</sup>))
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(≈3,67·10<sup>58</sup>,&nbsp;≈3,67·10<sup>58</sup>),<br>F<sub>1</sub>(≈3,67·10<sup>58</sup>,&nbsp;≈3,67·10<sup>58</sup>))
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(≈5,02·10<sup>135</sup>,&nbsp;≈5,02·10<sup>135</sup>),<br>F<sub>1</sub>(≈5,02·10<sup>135</sup>,&nbsp;≈5,02·10<sup>135</sup>))
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(≈1,43·10<sup>309</sup>,&nbsp;≈1,43·10<sup>309</sup>),<br>F<sub>1</sub>(≈1,43·10<sup>309</sup>,&nbsp;≈1,43·10<sup>309</sup>))
| F<sub>1</sub>&nbsp;(F<sub>1</sub>(≈3,36·10<sup>694</sup>,&nbsp;≈3,36·10<sup>694</sup>),<br>F<sub>1</sub>(≈3,36·10<sup>694</sup>,&nbsp;≈3,36·10<sup>694</sup>))
|-
| F<sub>1</sub>(88080360,<br>88080364) || F<sub>1</sub>(10230·2<sup>10231</sup>−10233,<br>10230·2<sup>10231</sup>−10229) || colspan="6" style="border:0; background:white" | 
|-
| ≈ 3,5&nbsp;·&nbsp;10<sup>26514839</sup> || colspan="7" style="border:0; background:white" | 
|-
| colspan="8" align="center" style="background:#FAF7F4" | much longer expression, starts with 2<sup>2<sup>2<sup>2<sup>x+1</sup></sup></sup></sup> an,  ≈ 10<sup>10<sup>10<sup>10<sup>lg&nbsp;2·(x+1) + lg(x+2)</sup></sup></sup></sup>
|}

=== F<sub>3</sub>의 값 ===
{| class="wikitable" style="text-align:right; font-size:90%"
|-
! <big>''<sub>y</sub>''&nbsp;\&nbsp;''<sup>x</sup>''</big> !! 0 !! 1 !! 2 !! 3 !! 4
|-
! rowspan="2" | 0
| 0 || 1 || 2 || 3 || 4
|-
| colspan="5" align="center" style="background:#FAF7F4" | ''x''
|-
! rowspan="4" | 1
| F<sub>2</sub>&nbsp;(F<sub>3</sub>(0,&nbsp;0), &nbsp;<br>F<sub>3</sub>(0,&nbsp;0)+1)
| F<sub>2</sub>&nbsp;(F<sub>3</sub>(1,&nbsp;0), &nbsp;<br>F<sub>3</sub>(1,&nbsp;0)+1)
| F<sub>2</sub>&nbsp;(F<sub>3</sub>(2,&nbsp;0), &nbsp;<br>F<sub>3</sub>(2,&nbsp;0)+1)
| F<sub>2</sub>&nbsp;(F<sub>3</sub>(3,&nbsp;0), &nbsp;<br>F<sub>3</sub>(3,&nbsp;0)+1)
| F<sub>2</sub>&nbsp;(F<sub>3</sub>(4,&nbsp;0), &nbsp;<br>F<sub>3</sub>(4,&nbsp;0)+1)
|-
| F<sub>2</sub>(0,&nbsp;1)
| F<sub>2</sub>(1,&nbsp;2)
| F<sub>2</sub>(2,&nbsp;3)
| F<sub>2</sub>(3,&nbsp;4)
| F<sub>2</sub>(4,&nbsp;5)
|-
| 1
| 10228
| ≈ 7,82&nbsp;·&nbsp;10<sup>4686813201</sup>
| style="background:white;" colspan="2" |
|-
| colspan="5" align="center" style="background:#FAF7F4" | No closed expressions possible within the framework of normal mathematical notation
|-
! rowspan="4" | 2
| F<sub>3</sub>&nbsp;(F<sub>4</sub>(0,&nbsp;1), &nbsp;<br>F<sub>4</sub>(0,&nbsp;1)+2)
| F<sub>3</sub>&nbsp;(F<sub>4</sub>(1,&nbsp;1), &nbsp;<br>F<sub>4</sub>(1,&nbsp;1)+2)
| F<sub>3</sub>&nbsp;(F<sub>4</sub>(2,&nbsp;1), &nbsp;<br>F<sub>4</sub>(2,&nbsp;1)+2)
| F<sub>3</sub>&nbsp;(F<sub>4</sub>(3,&nbsp;1), &nbsp;<br>F<sub>4</sub>(3,&nbsp;1)+2)
| F<sub>3</sub>&nbsp;(F<sub>4</sub>(4,&nbsp;1), &nbsp;<br>F<sub>4</sub>(4,&nbsp;1)+2)
|-
| F<sub>3</sub>&nbsp;(1,&nbsp;3)
| F<sub>3</sub>&nbsp;(10228,&nbsp;10230)
| F<sub>3</sub>&nbsp;(≈10<sup>4686813201</sup>,&nbsp;<br>≈10<sup>4686813201</sup>)
| style="background:white;" colspan="2" |
|-
| style="background:white;" colspan="5" | &nbsp;
|-
| colspan="5" align="center" style="background:#FAF7F4" | No closed expressions possible within the framework of normal mathematical notation
|}

== 외부 링크 ==
* {{웹 인용
  | author= [[Rosetta Code]]
  | title= Sudan function
  | url= https://rosettacode.org/wiki/Sudan_function
}}
* OEIS: [[oeis:A260003|A260003]], [[oeis:A260004|A260004]]
[[분류:산술]]
[[분류:특수 함수]]
[[분류:계산 이론]]