\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch1}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull1}{IntegerNumberTheoryFunctionsXmpPageEmpty1}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{div144 := divisors(144)\bound{div144 }}
\indentrel{3}\begin{verbatim}
   (1)  [1,2,3,4,6,8,9,12,16,18,24,36,48,72,144]
                                     Type: List Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty1}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty1}{IntegerNumberTheoryFunctionsXmpPagePatch1}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{div144 := divisors(144)\bound{div144 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch2}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull2}{IntegerNumberTheoryFunctionsXmpPageEmpty2}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{\#(div144)\free{div144 }}
\indentrel{3}\begin{verbatim}
   (2)  15
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty2}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty2}{IntegerNumberTheoryFunctionsXmpPagePatch2}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{\#(div144)\free{div144 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch3}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull3}{IntegerNumberTheoryFunctionsXmpPageEmpty3}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{reduce(+,div144)\free{div144 }}
\indentrel{3}\begin{verbatim}
   (3)  403
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty3}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty3}{IntegerNumberTheoryFunctionsXmpPagePatch3}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{reduce(+,div144)\free{div144 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch4}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull4}{IntegerNumberTheoryFunctionsXmpPageEmpty4}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{numberOfDivisors(144)}
\indentrel{3}\begin{verbatim}
   (4)  15
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty4}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty4}{IntegerNumberTheoryFunctionsXmpPagePatch4}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{numberOfDivisors(144)}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch5}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull5}{IntegerNumberTheoryFunctionsXmpPageEmpty5}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{sumOfDivisors(144)}
\indentrel{3}\begin{verbatim}
   (5)  403
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty5}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty5}{IntegerNumberTheoryFunctionsXmpPagePatch5}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{sumOfDivisors(144)}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch6}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull6}{IntegerNumberTheoryFunctionsXmpPageEmpty6}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f1 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty6}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty6}{IntegerNumberTheoryFunctionsXmpPagePatch6}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f1 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch7}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull7}{IntegerNumberTheoryFunctionsXmpPageEmpty7}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{f1(200)\free{f1 }}
\indentrel{3}\begin{verbatim}
   (7)  1
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty7}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty7}{IntegerNumberTheoryFunctionsXmpPagePatch7}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{f1(200)\free{f1 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch8}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull8}{IntegerNumberTheoryFunctionsXmpPageEmpty8}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{f1(846)\free{f1 }}
\indentrel{3}\begin{verbatim}
   (8)  1
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty8}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty8}{IntegerNumberTheoryFunctionsXmpPagePatch8}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{f1(846)\free{f1 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch9}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull9}{IntegerNumberTheoryFunctionsXmpPageEmpty9}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f2 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty9}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty9}{IntegerNumberTheoryFunctionsXmpPagePatch9}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f2 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch10}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull10}{IntegerNumberTheoryFunctionsXmpPageEmpty10}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{f2(200)\free{f2 }}
\indentrel{3}\begin{verbatim}
   (10)  200
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty10}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty10}{IntegerNumberTheoryFunctionsXmpPagePatch10}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{f2(200)\free{f2 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch11}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull11}{IntegerNumberTheoryFunctionsXmpPageEmpty11}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{f2(846)\free{f2 }}
\indentrel{3}\begin{verbatim}
   (11)  846
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty11}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty11}{IntegerNumberTheoryFunctionsXmpPagePatch11}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{f2(846)\free{f2 }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch12}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull12}{IntegerNumberTheoryFunctionsXmpPageEmpty12}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{fibonacci(25)}
\indentrel{3}\begin{verbatim}
   (12)  75025
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty12}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty12}{IntegerNumberTheoryFunctionsXmpPagePatch12}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{fibonacci(25)}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch13}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull13}{IntegerNumberTheoryFunctionsXmpPageEmpty13}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{[fibonacci(n) for n in 1..15]}
\indentrel{3}\begin{verbatim}
   (13)  [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610]
                                     Type: List Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty13}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty13}{IntegerNumberTheoryFunctionsXmpPagePatch13}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{[fibonacci(n) for n in 1..15]}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch14}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull14}{IntegerNumberTheoryFunctionsXmpPageEmpty14}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)])\bound{fib }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty14}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty14}{IntegerNumberTheoryFunctionsXmpPagePatch14}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)])\bound{fib }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch15}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull15}{IntegerNumberTheoryFunctionsXmpPageEmpty15}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{fib(25)\free{fib }}
\indentrel{3}\begin{verbatim}
   (15)  75025
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty15}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty15}{IntegerNumberTheoryFunctionsXmpPagePatch15}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{fib(25)\free{fib }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch16}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull16}{IntegerNumberTheoryFunctionsXmpPageEmpty16}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{[fib(n) for n in 1..15]\free{fib }}
\indentrel{3}\begin{verbatim}
   (16)  [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610]
                                     Type: List Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty16}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty16}{IntegerNumberTheoryFunctionsXmpPagePatch16}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{[fib(n) for n in 1..15]\free{fib }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch17}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull17}{IntegerNumberTheoryFunctionsXmpPageEmpty17}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull17}{\hidepaste}
\tab{5}\spadcommand{legendre(3,5)}
\indentrel{3}\begin{verbatim}
   (17)  - 1
                                          Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty17}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty17}{IntegerNumberTheoryFunctionsXmpPagePatch17}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty17}{\showpaste}
\tab{5}\spadcommand{legendre(3,5)}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch18}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull18}{IntegerNumberTheoryFunctionsXmpPageEmpty18}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull18}{\hidepaste}
\tab{5}\spadcommand{legendre(23,691)}
\indentrel{3}\begin{verbatim}
   (18)  - 1
                                          Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty18}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty18}{IntegerNumberTheoryFunctionsXmpPagePatch18}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty18}{\showpaste}
\tab{5}\spadcommand{legendre(23,691)}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch19}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull19}{IntegerNumberTheoryFunctionsXmpPageEmpty19}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull19}{\hidepaste}
\tab{5}\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2))\bound{h }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty19}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty19}{IntegerNumberTheoryFunctionsXmpPagePatch19}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty19}{\showpaste}
\tab{5}\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2))\bound{h }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch20}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull20}{IntegerNumberTheoryFunctionsXmpPageEmpty20}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull20}{\hidepaste}
\tab{5}\spadcommand{h(-163)\free{h }}
\indentrel{3}\begin{verbatim}
   (20)  1
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty20}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty20}{IntegerNumberTheoryFunctionsXmpPagePatch20}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty20}{\showpaste}
\tab{5}\spadcommand{h(-163)\free{h }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch21}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull21}{IntegerNumberTheoryFunctionsXmpPageEmpty21}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull21}{\hidepaste}
\tab{5}\spadcommand{h(-499)\free{h }}
\indentrel{3}\begin{verbatim}
   (21)  3
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty21}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty21}{IntegerNumberTheoryFunctionsXmpPagePatch21}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty21}{\showpaste}
\tab{5}\spadcommand{h(-499)\free{h }}
\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch22}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull22}{IntegerNumberTheoryFunctionsXmpPageEmpty22}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull22}{\hidepaste}
\tab{5}\spadcommand{h(-1832)\free{h }}
\indentrel{3}\begin{verbatim}
   (22)  26
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty22}
\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty22}{IntegerNumberTheoryFunctionsXmpPagePatch22}
\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty22}{\showpaste}
\tab{5}\spadcommand{h(-1832)\free{h }}
\end{paste}\end{patch}