1\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch1}
2\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull1}{IntegerNumberTheoryFunctionsXmpPageEmpty1}
3\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull1}{\hidepaste}
4\tab{5}\spadcommand{div144 := divisors(144)\bound{div144 }}
5\indentrel{3}\begin{verbatim}
6   (1)
7   [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144]
8                                    Type: List(Integer)
9\end{verbatim}
10\indentrel{-3}\end{paste}\end{patch}
11
12\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty1}
13\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty1}{IntegerNumberTheoryFunctionsXmpPagePatch1}
14\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty1}{\showpaste}
15\tab{5}\spadcommand{div144 := divisors(144)\bound{div144 }}
16\end{paste}\end{patch}
17
18\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch2}
19\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull2}{IntegerNumberTheoryFunctionsXmpPageEmpty2}
20\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull2}{\hidepaste}
21\tab{5}\spadcommand{\#(div144)\free{div144 }}
22\indentrel{3}\begin{verbatim}
23   (2)  15
24                                  Type: PositiveInteger
25\end{verbatim}
26\indentrel{-3}\end{paste}\end{patch}
27
28\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty2}
29\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty2}{IntegerNumberTheoryFunctionsXmpPagePatch2}
30\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty2}{\showpaste}
31\tab{5}\spadcommand{\#(div144)\free{div144 }}
32\end{paste}\end{patch}
33
34\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch3}
35\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull3}{IntegerNumberTheoryFunctionsXmpPageEmpty3}
36\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull3}{\hidepaste}
37\tab{5}\spadcommand{reduce(+,div144)\free{div144 }}
38\indentrel{3}\begin{verbatim}
39   (3)  403
40                                  Type: PositiveInteger
41\end{verbatim}
42\indentrel{-3}\end{paste}\end{patch}
43
44\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty3}
45\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty3}{IntegerNumberTheoryFunctionsXmpPagePatch3}
46\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty3}{\showpaste}
47\tab{5}\spadcommand{reduce(+,div144)\free{div144 }}
48\end{paste}\end{patch}
49
50\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch4}
51\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull4}{IntegerNumberTheoryFunctionsXmpPageEmpty4}
52\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull4}{\hidepaste}
53\tab{5}\spadcommand{numberOfDivisors(144)}
54\indentrel{3}\begin{verbatim}
55   (4)  15
56                                  Type: PositiveInteger
57\end{verbatim}
58\indentrel{-3}\end{paste}\end{patch}
59
60\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty4}
61\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty4}{IntegerNumberTheoryFunctionsXmpPagePatch4}
62\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty4}{\showpaste}
63\tab{5}\spadcommand{numberOfDivisors(144)}
64\end{paste}\end{patch}
65
66\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch5}
67\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull5}{IntegerNumberTheoryFunctionsXmpPageEmpty5}
68\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull5}{\hidepaste}
69\tab{5}\spadcommand{sumOfDivisors(144)}
70\indentrel{3}\begin{verbatim}
71   (5)  403
72                                  Type: PositiveInteger
73\end{verbatim}
74\indentrel{-3}\end{paste}\end{patch}
75
76\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty5}
77\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty5}{IntegerNumberTheoryFunctionsXmpPagePatch5}
78\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty5}{\showpaste}
79\tab{5}\spadcommand{sumOfDivisors(144)}
80\end{paste}\end{patch}
81
82\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch6}
83\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull6}{IntegerNumberTheoryFunctionsXmpPageEmpty6}
84\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull6}{\hidepaste}
85\tab{5}\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f1 }}
86\indentrel{3}\begin{verbatim}
87                                             Type: Void
88\end{verbatim}
89\indentrel{-3}\end{paste}\end{patch}
90
91\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty6}
92\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty6}{IntegerNumberTheoryFunctionsXmpPagePatch6}
93\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty6}{\showpaste}
94\tab{5}\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f1 }}
95\end{paste}\end{patch}
96
97\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch7}
98\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull7}{IntegerNumberTheoryFunctionsXmpPageEmpty7}
99\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull7}{\hidepaste}
100\tab{5}\spadcommand{f1(200)\free{f1 }}
101\indentrel{3}\begin{verbatim}
102   (7)  1
103                                  Type: PositiveInteger
104\end{verbatim}
105\indentrel{-3}\end{paste}\end{patch}
106
107\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty7}
108\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty7}{IntegerNumberTheoryFunctionsXmpPagePatch7}
109\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty7}{\showpaste}
110\tab{5}\spadcommand{f1(200)\free{f1 }}
111\end{paste}\end{patch}
112
113\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch8}
114\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull8}{IntegerNumberTheoryFunctionsXmpPageEmpty8}
115\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull8}{\hidepaste}
116\tab{5}\spadcommand{f1(846)\free{f1 }}
117\indentrel{3}\begin{verbatim}
118   (8)  1
119                                  Type: PositiveInteger
120\end{verbatim}
121\indentrel{-3}\end{paste}\end{patch}
122
123\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty8}
124\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty8}{IntegerNumberTheoryFunctionsXmpPagePatch8}
125\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty8}{\showpaste}
126\tab{5}\spadcommand{f1(846)\free{f1 }}
127\end{paste}\end{patch}
128
129\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch9}
130\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull9}{IntegerNumberTheoryFunctionsXmpPageEmpty9}
131\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull9}{\hidepaste}
132\tab{5}\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f2 }}
133\indentrel{3}\begin{verbatim}
134                                             Type: Void
135\end{verbatim}
136\indentrel{-3}\end{paste}\end{patch}
137
138\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty9}
139\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty9}{IntegerNumberTheoryFunctionsXmpPagePatch9}
140\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty9}{\showpaste}
141\tab{5}\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f2 }}
142\end{paste}\end{patch}
143
144\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch10}
145\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull10}{IntegerNumberTheoryFunctionsXmpPageEmpty10}
146\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull10}{\hidepaste}
147\tab{5}\spadcommand{f2(200)\free{f2 }}
148\indentrel{3}\begin{verbatim}
149   (10)  200
150                                  Type: PositiveInteger
151\end{verbatim}
152\indentrel{-3}\end{paste}\end{patch}
153
154\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty10}
155\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty10}{IntegerNumberTheoryFunctionsXmpPagePatch10}
156\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty10}{\showpaste}
157\tab{5}\spadcommand{f2(200)\free{f2 }}
158\end{paste}\end{patch}
159
160\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch11}
161\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull11}{IntegerNumberTheoryFunctionsXmpPageEmpty11}
162\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull11}{\hidepaste}
163\tab{5}\spadcommand{f2(846)\free{f2 }}
164\indentrel{3}\begin{verbatim}
165   (11)  846
166                                  Type: PositiveInteger
167\end{verbatim}
168\indentrel{-3}\end{paste}\end{patch}
169
170\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty11}
171\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty11}{IntegerNumberTheoryFunctionsXmpPagePatch11}
172\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty11}{\showpaste}
173\tab{5}\spadcommand{f2(846)\free{f2 }}
174\end{paste}\end{patch}
175
176\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch12}
177\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull12}{IntegerNumberTheoryFunctionsXmpPageEmpty12}
178\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull12}{\hidepaste}
179\tab{5}\spadcommand{fibonacci(25)}
180\indentrel{3}\begin{verbatim}
181   (12)  75025
182                                  Type: PositiveInteger
183\end{verbatim}
184\indentrel{-3}\end{paste}\end{patch}
185
186\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty12}
187\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty12}{IntegerNumberTheoryFunctionsXmpPagePatch12}
188\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty12}{\showpaste}
189\tab{5}\spadcommand{fibonacci(25)}
190\end{paste}\end{patch}
191
192\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch13}
193\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull13}{IntegerNumberTheoryFunctionsXmpPageEmpty13}
194\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull13}{\hidepaste}
195\tab{5}\spadcommand{[fibonacci(n) for n in 1..15]}
196\indentrel{3}\begin{verbatim}
197   (13)
198   [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
199    610]
200                                    Type: List(Integer)
201\end{verbatim}
202\indentrel{-3}\end{paste}\end{patch}
203
204\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty13}
205\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty13}{IntegerNumberTheoryFunctionsXmpPagePatch13}
206\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty13}{\showpaste}
207\tab{5}\spadcommand{[fibonacci(n) for n in 1..15]}
208\end{paste}\end{patch}
209
210\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch14}
211\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull14}{IntegerNumberTheoryFunctionsXmpPageEmpty14}
212\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull14}{\hidepaste}
213\tab{5}\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)])\bound{fib }}
214\indentrel{3}\begin{verbatim}
215                                             Type: Void
216\end{verbatim}
217\indentrel{-3}\end{paste}\end{patch}
218
219\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty14}
220\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty14}{IntegerNumberTheoryFunctionsXmpPagePatch14}
221\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty14}{\showpaste}
222\tab{5}\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)])\bound{fib }}
223\end{paste}\end{patch}
224
225\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch15}
226\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull15}{IntegerNumberTheoryFunctionsXmpPageEmpty15}
227\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull15}{\hidepaste}
228\tab{5}\spadcommand{fib(25)\free{fib }}
229\indentrel{3}\begin{verbatim}
230   (15)  75025
231                                  Type: PositiveInteger
232\end{verbatim}
233\indentrel{-3}\end{paste}\end{patch}
234
235\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty15}
236\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty15}{IntegerNumberTheoryFunctionsXmpPagePatch15}
237\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty15}{\showpaste}
238\tab{5}\spadcommand{fib(25)\free{fib }}
239\end{paste}\end{patch}
240
241\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch16}
242\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull16}{IntegerNumberTheoryFunctionsXmpPageEmpty16}
243\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull16}{\hidepaste}
244\tab{5}\spadcommand{[fib(n) for n in 1..15]\free{fib }}
245\indentrel{3}\begin{verbatim}
246   (16)
247   [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
248    610]
249                                    Type: List(Integer)
250\end{verbatim}
251\indentrel{-3}\end{paste}\end{patch}
252
253\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty16}
254\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty16}{IntegerNumberTheoryFunctionsXmpPagePatch16}
255\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty16}{\showpaste}
256\tab{5}\spadcommand{[fib(n) for n in 1..15]\free{fib }}
257\end{paste}\end{patch}
258
259\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch17}
260\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull17}{IntegerNumberTheoryFunctionsXmpPageEmpty17}
261\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull17}{\hidepaste}
262\tab{5}\spadcommand{legendre(3,5)}
263\indentrel{3}\begin{verbatim}
264   (17)  - 1
265                                          Type: Integer
266\end{verbatim}
267\indentrel{-3}\end{paste}\end{patch}
268
269\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty17}
270\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty17}{IntegerNumberTheoryFunctionsXmpPagePatch17}
271\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty17}{\showpaste}
272\tab{5}\spadcommand{legendre(3,5)}
273\end{paste}\end{patch}
274
275\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch18}
276\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull18}{IntegerNumberTheoryFunctionsXmpPageEmpty18}
277\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull18}{\hidepaste}
278\tab{5}\spadcommand{legendre(23,691)}
279\indentrel{3}\begin{verbatim}
280   (18)  - 1
281                                          Type: Integer
282\end{verbatim}
283\indentrel{-3}\end{paste}\end{patch}
284
285\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty18}
286\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty18}{IntegerNumberTheoryFunctionsXmpPagePatch18}
287\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty18}{\showpaste}
288\tab{5}\spadcommand{legendre(23,691)}
289\end{paste}\end{patch}
290
291\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch19}
292\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull19}{IntegerNumberTheoryFunctionsXmpPageEmpty19}
293\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull19}{\hidepaste}
294\tab{5}\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2))\bound{h }}
295\indentrel{3}\begin{verbatim}
296                                             Type: Void
297\end{verbatim}
298\indentrel{-3}\end{paste}\end{patch}
299
300\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty19}
301\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty19}{IntegerNumberTheoryFunctionsXmpPagePatch19}
302\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty19}{\showpaste}
303\tab{5}\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2))\bound{h }}
304\end{paste}\end{patch}
305
306\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch20}
307\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull20}{IntegerNumberTheoryFunctionsXmpPageEmpty20}
308\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull20}{\hidepaste}
309\tab{5}\spadcommand{h(-163)\free{h }}
310\indentrel{3}\begin{verbatim}
311   (20)  1
312                                  Type: PositiveInteger
313\end{verbatim}
314\indentrel{-3}\end{paste}\end{patch}
315
316\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty20}
317\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty20}{IntegerNumberTheoryFunctionsXmpPagePatch20}
318\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty20}{\showpaste}
319\tab{5}\spadcommand{h(-163)\free{h }}
320\end{paste}\end{patch}
321
322\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch21}
323\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull21}{IntegerNumberTheoryFunctionsXmpPageEmpty21}
324\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull21}{\hidepaste}
325\tab{5}\spadcommand{h(-499)\free{h }}
326\indentrel{3}\begin{verbatim}
327   (21)  3
328                                  Type: PositiveInteger
329\end{verbatim}
330\indentrel{-3}\end{paste}\end{patch}
331
332\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty21}
333\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty21}{IntegerNumberTheoryFunctionsXmpPagePatch21}
334\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty21}{\showpaste}
335\tab{5}\spadcommand{h(-499)\free{h }}
336\end{paste}\end{patch}
337
338\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch22}
339\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull22}{IntegerNumberTheoryFunctionsXmpPageEmpty22}
340\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull22}{\hidepaste}
341\tab{5}\spadcommand{h(-1832)\free{h }}
342\indentrel{3}\begin{verbatim}
343   (22)  26
344                                  Type: PositiveInteger
345\end{verbatim}
346\indentrel{-3}\end{paste}\end{patch}
347
348\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty22}
349\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty22}{IntegerNumberTheoryFunctionsXmpPagePatch22}
350\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty22}{\showpaste}
351\tab{5}\spadcommand{h(-1832)\free{h }}
352\end{paste}\end{patch}
353
354