# How to automatically draw tree diagram of prime factorization with LaTeX?

I want to have a simple interface to automatically draw tree diagrams of prime factorization. For example, by invoking the following code,

\PrimeTree{36}
\PrimeTree{90}
\PrimeTree{112}
\PrimeTree{612}
\PrimeTree{7875}
\PrimeTree{22230}


we will get the following output.

How to do this in LaTeX (PSTricks, TikZ, Asymptote, Metapost, etc)?

My uneducated effort is as follows.

\documentclass[border=3pt,preview,varwidth,multi]{standalone}
\usepackage{pst-tree}
\psset{levelsep=1,treesep=1,nodesep=2pt}

\begin{document}

\preview
\psTree{\TR{36}}
\Tcircle{2}
\psTree{\TR{18}}
\Tcircle{2}
\psTree{\TR{9}}
\Tcircle{3}
\Tcircle{3}
\endpsTree
\endpsTree
\endpsTree
\endpreview

\preview
\psTree{\TR{90}}
\Tcircle{2}
\psTree{\TR{45}}
\Tcircle{3}
\psTree{\TR{15}}
\Tcircle{3}
\Tcircle{5}
\endpsTree
\endpsTree
\endpsTree
\endpreview

\preview
\psTree{\TR{112}}
\Tcircle{2}
\psTree{\TR{56}}
\Tcircle{2}
\psTree{\TR{28}}
\Tcircle{2}
\psTree{\TR{14}}
\Tcircle{2}
\Tcircle{7}
\endpsTree
\endpsTree
\endpsTree
\endpsTree
\endpreview

\preview
\psTree{\TR{612}}
\Tcircle{2}
\psTree{\TR{306}}
\Tcircle{2}
\psTree{\TR{153}}
\Tcircle{3}
\psTree{\TR{51}}
\Tcircle{3}
\Tcircle{17}
\endpsTree
\endpsTree
\endpsTree
\endpsTree
\endpreview

\preview
\psTree{\TR{7875}}
\Tcircle{3}
\psTree{\TR{2625}}
\Tcircle{3}
\psTree{\TR{875}}
\Tcircle{5}
\psTree{\TR{175}}
\Tcircle{5}
\psTree{\TR{35}}
\Tcircle{5}
\Tcircle{7}
\endpsTree
\endpsTree
\endpsTree
\endpsTree
\endpsTree
\endpreview

\preview
\psTree{\TR{22230}}
\Tcircle{2}
\psTree{\TR{11115}}
\Tcircle{3}
\psTree{\TR{3705}}
\Tcircle{3}
\psTree{\TR{1235}}
\Tcircle{5}
\psTree{\TR{247}}
\Tcircle{13}
\Tcircle{19}
\endpsTree
\endpsTree
\endpsTree
\endpsTree
\endpsTree
\endpreview
\end{document}


In order to appreciate your effort, I can only offer the following!

-
How can you offer 4 bounties of 500 each with a reputation of 1k? – gerrit Sep 5 '13 at 18:13
@gerrit don't worry, PSTikZ is a master of bounties :) – cmhughes Sep 5 '13 at 20:14
Hm, I see. – gerrit Sep 5 '13 at 20:20
On the plus side, you've also given Qrrbrbirlbel a gold badge by accepting the other answer :) – cmhughes Sep 6 '13 at 17:09
@PSTikZ I have now written, based on the \xintGCD command of package xintgcd, (non-optimized) macros \GCDof and \LCMof, for example \LCMof {{5549544000}{2499792094800}{2473511040}{416215800}} gives 99991683792000. – jfbu Sep 6 '13 at 19:43

## 8 Answers

This answer has evolved in stages. New contributions were added at the bottom (apart from the images which I put on top).

1. macro \factorize to produce programmatically the factorization of input N. The output is put in macro \factors. For example for N=36, \factors expands to {{}{}{36}}{{2}{2}{9}}{{3}{2}{1}}. Each successive triplet is {p}{k}{m} where p^k is the p-factor of N, and m the result of dividing N by it and all previous ones. The first triple {}{}{N} is a bit of a nuisance, and this explains \expandafter\@gobble\factors in some the displaying codes. Initially the displays used tabulars.

2. then I added code using pst-tree to produce the trees from the result available in \factors. I also added code for just printing the factorization inline.

3. I now add code using TikZ+forest, which I have learned a bit since seeing it used beautifully in Qrrbrbirlbel's answer. The code for the forest tree also incoporates the inline version of the factorization.

I use pst-tree and forest only to display, not to compute the factorization. Both pst-tree and forest syntax for the trees allowed a simple approach to work from \factors to the trees: two token lists are filled-up step by step; if using the native TikZ syntax with child, that would be much more difficult because of braces { and }.

Images of trees produced with forest:

Images of trees using pst-tree (as were done manually by PSTikZ.) Here are some samples (the code also has the variant to not display the 1's when they are exponents). For some other ways to do trees, e.g with TikZ's childs and nodes, the \factors macro prepared by the \factorize command should probably be done in a slightly different manner.

I too propose half of an answer using package xint for the arithmetic computations (on arbitrarily big numbers). The command \factorize{N} results in macro \factors containing a list of triplets {p}{k}{m}, where p is a prime number showing up in the factorization, k is its exponent, m is the result of division of N by p^k as well as all previous factors corresponding to smaller primes. So the last triplet has m=1, and the first is {}{}{N}.

The tree could then be constructed from this macro \factors; here I just have a \displayfactorization to display the result using tabulars (and I use some macros of xint to transform \factors into what goes into the tabular). There is an optional argument to set up the width of the second column.

I use the numprint package to print very long numbers without going beyond the page physical limits.

The algorithm is very lame: first divisions by 2 are tested, then 3, then 5, and all successive odd integers until N has been reduced to 1.

There is no impact on TeX memory, apart from \factors being filled up.

But 2147483647 would take a very long time (I did not estimate).

UPDATE: I was too pessimistic for 2147483647; just need to be a bit patient... turns out to take about 8 seconds on my 2012 laptop. see added examples at the bottom.

UPDATE 2: now also tested on 19928968819, here is the output from using \factorizeanddisplay{19928968819} as defined at the bottom of this answer:

The algorithm would be of course faster if done using only TeX \count registers (and eTeX \numexpr for convenience), but this would restrict it to numbers < 2^{31}.

The code is a bit obscure in particular because the package does not (*) provide a convenient public test for a number being 1 (it has one for testing for equality with zero, but subtracting 1 would be a loss of efficiency), and I had to use a non-public macro from the package source together with some \expandafter's.

(*) I had a bit forgotten the package commands; \xintGeq{\x}{2} or \xintCmp{\x}{1} both return 1 if \x is an integer >1, and 0 if \x=1 so I could have used that rather than the more private and cumbersome (it does not expand its argument) \XINT_isOne (which is faster, though, so I maintain its use in the code).

update: I have incorporated the usual halting test to not go beyond square root of n, makes the thing a bit more efficient when the factorization ends in a 'big' prime (two such examples added).

\documentclass{article}
\usepackage{xint}
\usepackage{xintexpr}
\usepackage[T1]{fontenc}
\usepackage{array}
\usepackage[np]{numprint}
\AtBeginDocument{\npthousandsep{,\hskip .1pt plus .02pt minus .02pt}}

\catcode\_ 11

\def\factorize#1{%
\edef\factorize_N{#1}%
\def\factorize_exp{0}%
\edef\factors{{{}{}{\factorize_N}}}%
\factorize_i
}

\def\factorize_i{%
\if\xintOdd{\factorize_N}1%
\expandafter \factorize_ii
\else
\edef\factorize_exp{\xintInc{\factorize_exp}}%
\edef\factorize_N{\xintHalf{\factorize_N}}%
\expandafter \factorize_i
\fi
}

\def\factorize_ii{%
\if\xintSgn{\factorize_exp}1%
\edef\factors{\factors{{2}{\factorize_exp}{\factorize_N}}}%
\fi
\if\expandafter\XINT_isOne\expandafter{\factorize_N}1%
\else
\def\factorize_M{3}%
\def\factorize_exp{0}%
\expandafter \factorize_iii
\fi
}

\def\factorize_iii{%
\xintAssign\xintDivision\factorize_N\factorize_M\to
\factorize_Q\factorize_R
\xintSgnFork{\xintSgn\factorize_R}%
{}%
{\edef\factorize_exp{\xintInc{\factorize_exp}}%
\let\factorize_N\factorize_Q
\factorize_iii}%
{\factorize_iv}%
}

\def\factorize_iv{%
\if\xintSgn{\factorize_exp}1%
\edef\factors{\factors{{\factorize_M}{\factorize_exp}{\factorize_N}}}%
\fi
\if\expandafter\XINT_isOne\expandafter{\factorize_N}1%
\else
% here N>1, N=QM+R (0<R<M) is < M(Q+1) and N has no prime factors
% at most equal to M. If a prime P>M divides N, the
% quotient N/P will be < Q+1, hence at most Q. If Q<=M, then
% N/P must be 1 else there would be some prime <=M dividing N.
\if\xintGeq\factorize_M\factorize_Q 1% implies that N is prime
\edef\factors{\factors{{\factorize_N}{1}{1}}}% we stop here
\else % we go on testing with bigger factors
% \edef\factorize_M{\xintInc{\xintInc{\factorize_M}}}%
\edef\factorize_M{\xintiAdd \factorize_M 2}%
\def\factorize_exp{0}%
\expandafter \expandafter \expandafter \factorize_iii
\fi
\fi
}

\catcode\_ 8

\def\auxiliarymacroa #1{\auxiliarymacrob #1}
\def\auxiliarymacrob #1#2#3{${#1}^{#2}$&\np{#3}\tabularnewline\hline}

\newcommand*\displayfactorization[1][.2\linewidth]{%
\begin{tabular}{>{\rule{0pt}{11pt}}c|>{\raggedright}p{#1}}%
\xintApplyUnbraced\auxiliarymacroa{\factors}
\end{tabular}\hskip.5em plus .125em minus .125em }

%for testing:
% \def\displayfactorization{\meaning\factors}

\pagestyle{empty}
\begin{document}\thispagestyle{empty}\ttfamily

\noindent\factorize{36}\displayfactorization
\factorize{90}\displayfactorization
\factorize{112}\displayfactorization
\factorize{612}\displayfactorization
\factorize{7875}\displayfactorization
\factorize{22230}\displayfactorization
\factorize{1073741824}\displayfactorization
\factorize{2147483644}\displayfactorization
\factorize{\xintiPow 2{40}}\displayfactorization
\factorize{\xintNum{\xinttheexpr 2^{37}*3^{23}*17^{13}\relax}}%
\displayfactorization
% two examples ending with a (somewhat) big' prime,
\factorize{10968058712}\displayfactorization
\factorize{1689242184972}\displayfactorization

\factorize{\xintNum{\xinttheexpr 111^{13}*371^{7}*1271^{35}\relax}}
\displayfactorization[.75\linewidth]

% \factorize{2147483647}% does not exhaust memory but will take much time it seems

\end{document}


and an example with big integers:

I was too pessimistic about 2147483647:

\def\factorizeanddisplay #1{%
\pdfresettimer
\factorize{#1}%
\edef\z{\the\pdfelapsedtime}%
(needed \xintRound{2}{\z/65536} seconds)
\displayfactorization
\par\medskip
}

\factorizeanddisplay{2147483642}
\factorizeanddisplay{2147483643}
\factorizeanddisplay{2147483644}
\factorizeanddisplay{2147483645}
\factorizeanddisplay{2147483646}
\factorizeanddisplay{2147483647}
\factorizeanddisplay{2147483648}
\factorizeanddisplay{2147483649}
\factorizeanddisplay{2147483650}


And now the code repeated, together with code to produce trees in the format of the OP question, using pst-tree:

\documentclass[border=3pt,preview,varwidth,multi]{standalone}

\usepackage{pst-tree}
\psset{levelsep=1,treesep=1,nodesep=2pt}

\usepackage{xint}
\usepackage{xintexpr}

\catcode\_ 11

% This code (non-expandable) produces
% {{}{}{N}} followed with successive braced triplets
% {{p}{k}{m}} where p is a prime factor of N,
% k its exponent in N, and m is the result of dividing
% N by p^k and all previous powers of smaller primes
% So the last triplet has m=1

% The code uses package xint to be able to deal
% with numbers larger than the TeX limit of 2^{31}-1
% on count registers.

\def\factorize#1{%
\edef\factorize_N{#1}%
\def\factorize_exp{0}%
\edef\factors{{{}{}{\factorize_N}}}%
\factorize_i
}

\def\factorize_i{%
\if\xintOdd{\factorize_N}1%
\expandafter \factorize_ii
\else
\edef\factorize_exp{\xintInc{\factorize_exp}}%
\edef\factorize_N{\xintHalf{\factorize_N}}%
\expandafter \factorize_i
\fi
}

\def\factorize_ii{%
\if\xintSgn{\factorize_exp}1%
\edef\factors{\factors{{2}{\factorize_exp}{\factorize_N}}}%
\fi
\if\expandafter\XINT_isOne\expandafter{\factorize_N}1%
\else
\def\factorize_M{3}%
\def\factorize_exp{0}%
\expandafter \factorize_iii
\fi
}

\def\factorize_iii{%
\xintAssign\xintDivision\factorize_N\factorize_M\to
\factorize_Q\factorize_R
\xintSgnFork{\xintSgn\factorize_R}%
{}%
{\edef\factorize_exp{\xintInc{\factorize_exp}}%
\let\factorize_N\factorize_Q
\factorize_iii}%
{\factorize_iv}%
}

\def\factorize_iv{%
\if\xintSgn{\factorize_exp}1%
\edef\factors{\factors{{\factorize_M}{\factorize_exp}{\factorize_N}}}%
\fi
\if\expandafter\XINT_isOne\expandafter{\factorize_N}1%
\else
% here N>1, N=QM+R (0<R<Q) is < M(Q+1) and N has no prime factors
% at most equal to M. If a prime P>M divides N, the
% quotient N/P will be < Q+1, hence at most Q. If Q<=M, then
% N/P must be 1 else there would be some prime <=M dividing N.
\if\xintGeq\factorize_M\factorize_Q 1% implies that N is prime
\edef\factors{\factors{{\factorize_N}{1}{1}}}% we stop here
\else % we go on testing with bigger factors
% \edef\factorize_M{\xintInc{\xintInc{\factorize_M}}}%
\edef\factorize_M{\xintiAdd \factorize_M 2}%
\def\factorize_exp{0}%
\expandafter \expandafter \expandafter \factorize_iii
\fi
\fi
}

\catcode\_ 8

%% We now define the macro \FactorTree which will produce a tree
%% displaying the factorization, using PSTricks

\newtoks\FactorTreeA
\newtoks\FactorTreeB

\makeatletter

\newcommand*{\FactorsToTree}[1]{%
\FactorsToTree@ #1%
}

% macro which was used to produce the images; variant follows which skips the
% exponents equal to 1.

% \newcommand*{\FactorsToTree@}[3]{%
%     \xintSgnFork{\xintCmp{#3}{1}}% check to see if end has been reached
%     {}%
%     {\FactorTreeA\expandafter{\the\FactorTreeA
%                               \Tcircle{${#1}^{#2}$}%
%                               \TR{1}%
%                               }}%
%     {\FactorTreeA\expandafter{\the\FactorTreeA
%                              \Tcircle{${#1}^{#2}$}%
%                              \psTree{\TR{#3}}}%
%      \FactorTreeB\expandafter{\the\FactorTreeB \endpsTree}}%
% }

% This variant will not print the exponents equal to 1:

\newcommand*{\FactorsToTree@}[3]{%
\ifnum 0#2=1 % first triplet has an empty #2, hence the trick with 0
\expandafter\@firstoftwo
\else
\expandafter\@secondoftwo
\fi
% exponent #2 is 1, so don't print it
{\xintSgnFork{\xintCmp{#3}{1}}% check to see if end has been reached
{}%
{\FactorTreeA\expandafter{\the\FactorTreeA
\Tcircle{${#1}$}%
\TR{1}%
}}%
{\FactorTreeA\expandafter{\the\FactorTreeA
\Tcircle{${#1}$}%
\psTree{\TR{#3}}}%
\FactorTreeB\expandafter{\the\FactorTreeB \endpsTree}}}
% exponent #2 is > 1 (or absent in the {}{}{N} triplet)
{\xintSgnFork{\xintCmp{#3}{1}}% check to see if end has been reached
{}%
{\FactorTreeA\expandafter{\the\FactorTreeA
\Tcircle{${#1}^{#2}$}%
\TR{1}%
}}%
{\FactorTreeA\expandafter{\the\FactorTreeA
\Tcircle{${#1}^{#2}$}%
\psTree{\TR{#3}}}%
\FactorTreeB\expandafter{\the\FactorTreeB \endpsTree}}}%
}

\newcommand*{\FactorTree}[1]{%
\factorize{#1}%
\FactorTreeA{\@gobbletwo}%
\FactorTreeB{}%
\xintApplyUnbraced\FactorsToTree{\factors}%
\the\FactorTreeA\the\FactorTreeB
}

\makeatother

\begin{document}

%% \preview\FactorTree{1}\endpreview  %% (ok)

\preview\FactorTree{13}\endpreview

\preview\FactorTree{36}\endpreview

\preview\FactorTree{90}\endpreview

\preview\FactorTree{112}\endpreview

\preview\FactorTree{612}\endpreview

\preview\FactorTree{7875}\endpreview

\preview\FactorTree{22230}\endpreview

\preview\FactorTree{1073741824}\endpreview

\preview\FactorTree{2147483644}\endpreview

\preview\FactorTree{\xintiPow 2{40}}\endpreview

\preview\FactorTree{\xintNum{\xinttheexpr 2^{37}*3^{23}*17^{13}\relax}}%
\endpreview
% two examples ending with a (somewhat) big' prime,

\preview\FactorTree{10968058712}\endpreview

\preview\FactorTree{1689242184972}\endpreview

\end{document}


Code for inline product decomposition:

% The command \FactorizeInline produces (for math mode, at it uses \times) the
% product decomposition of its argument into prime powers, the exponents equal
% to 1 are not printed. The argument is supposed to be an integer > 1
% (arbitrarily big, but computation times will be large if it is not a product
% of reasonably small primes).
% $N = \FactorizeInline{N}$

\makeatletter
\def\@factorinliner  #1{\@factorinliner@ #1}
\def\@factorinliner@ #1#2#3{\ifnum #2>1 \expandafter\@firstoftwo\else
\expandafter\@secondoftwo\fi
{{#1}^{#2}}{#1}}
\newcommand*{\FactorizeInline}[1]{%
\factorize{#1}%
\xintListWithSep\times
{\xintApply\@factorinliner{\expandafter\@gobble\factors}}%
}%
\makeatother

$13=\FactorizeInline{13}$

$36=\FactorizeInline{36}$

$90=\FactorizeInline{90}$

$112=\FactorizeInline{112}$

$612=\FactorizeInline{612}$

$7875=\FactorizeInline{7875}$

$22230=\FactorizeInline{22230}$

$1073741824=\FactorizeInline{1073741824}$

$2147483642=\FactorizeInline{2147483642}$

% $2147483643=\FactorizeInline{2147483643}$ % 4.5 seconds on my laptop

$2147483644=\FactorizeInline{2147483644}$

$2147483645=\FactorizeInline{2147483645}$

% $2147483646=\FactorizeInline{2147483646}$

% $2147483647=\FactorizeInline{2147483647}$ % 8 seconds on my 2012 laptop

$2147483648=\FactorizeInline{2147483648}$

% $2147483649=\FactorizeInline{2147483649}$ % 4.5 seconds on my laptop

$2147483650=\FactorizeInline{2147483650}$

% $19928968819=\FactorizeInline{19928968819}$ % 25 seconds on my 2012 laptop

$\xintiPow 2{40} = \FactorizeInline{\xintiPow 2{40}}$

% two examples ending with a (somewhat) big' prime,

$10968058712=\FactorizeInline{10968058712}$

$1689242184972=\FactorizeInline{1689242184972}$

%\xintNum{\xinttheexpr 2^{37}*3^{23}*17^{13}\relax}

$128154740640622513993964746937443811328=\FactorizeInline{128154740640622513993964746937443811328}$


Code for doing the trees with forest:

\documentclass[border=3pt,varwidth,multi={forest}]{standalone}

\usepackage{calc} % for \widthof

\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{forest}

\usepackage{xint}
\usepackage{xintexpr}

%  macro \factorize as above
\catcode\_ 11

% This code (non-expandable) produces
% {{}{}{N}} followed with successive braced triplets
% {{p}{k}{m}} where p is a prime factor of N,
% k its exponent in N, and m is the result of dividing
% N by p^k and all previous powers of smaller primes
% So the last triplet has m=1

% The code uses package xint to be able to deal
% with numbers larger than the TeX limit of 2^{31}-1
% on count registers.

\def\factorize#1{%
\edef\factorize_N{#1}%
\def\factorize_exp{0}%
\edef\factors{{{}{}{\factorize_N}}}%
\factorize_i
}

\def\factorize_i{%
\if\xintOdd{\factorize_N}1%
\expandafter \factorize_ii
\else
\edef\factorize_exp{\xintInc{\factorize_exp}}%
\edef\factorize_N{\xintHalf{\factorize_N}}%
\expandafter \factorize_i
\fi
}

\def\factorize_ii{%
\if\xintSgn{\factorize_exp}1%
\edef\factors{\factors{{2}{\factorize_exp}{\factorize_N}}}%
\fi
\if\expandafter\XINT_isOne\expandafter{\factorize_N}1%
\else
\def\factorize_M{3}%
\def\factorize_exp{0}%
\expandafter \factorize_iii
\fi
}

\def\factorize_iii{%
\xintAssign\xintDivision\factorize_N\factorize_M\to
\factorize_Q\factorize_R
\xintSgnFork{\xintSgn\factorize_R}%
{}%
{\edef\factorize_exp{\xintInc{\factorize_exp}}%
\let\factorize_N\factorize_Q
\factorize_iii}%
{\factorize_iv}%
}

\def\factorize_iv{%
\if\xintSgn{\factorize_exp}1%
\edef\factors{\factors{{\factorize_M}{\factorize_exp}{\factorize_N}}}%
\fi
\if\expandafter\XINT_isOne\expandafter{\factorize_N}1%
\else
% here N>1, N=QM+R (0<R<Q) is < M(Q+1) and N has no prime factors
% at most equal to M. If a prime P>M divides N, the
% quotient N/P will be < Q+1, hence at most Q. If Q<=M, then
% N/P must be 1 else there would be some prime <=M dividing N.
\if\xintGeq\factorize_M\factorize_Q 1% implies that N is prime
\edef\factors{\factors{{\factorize_N}{1}{1}}}% we stop here
\else % we go on testing with bigger factors
% \edef\factorize_M{\xintInc{\xintInc{\factorize_M}}}%
\edef\factorize_M{\xintiAdd \factorize_M 2}%
\def\factorize_exp{0}%
\expandafter \expandafter \expandafter \factorize_iii
\fi
\fi
}

\catcode\_ 8

%% We now define the macro \FactorTree which will produce a tree
%% displaying the factorization,
%% using TikZ+forest (for the bracket syntax, much easier to deal with compared
%% to the braces-based child-node native TikZ tree syntax)

\makeatletter

\newtoks\FactorTreeA
\newtoks\FactorTreeB

\newcommand*{\FactorsToTree@}[3]{%
\ifnum #2=1 %
\expandafter\@firstoftwo
\else
\expandafter\@secondoftwo
\fi
% exponent #2 is 1, so don't print it
{\xintSgnFork{\xintCmp{#3}{1}}% check to see if end has been reached
{}%
% here, this is the last triplet and it has the shape {P}{1}{1}
% and P was already inserted as tree node in the previous step
% and the forest syntax allows to insert options here
{\FactorTreeA\expandafter{\the\FactorTreeA ,draw,circle}}%
{\FactorTreeA\expandafter{\the\FactorTreeA
[{${#1}$}]
[{${#3}$}%
}%
\FactorTreeB\expandafter{\the\FactorTreeB ]}%
}}%
% exponent #2 is > 1
{\xintSgnFork{\xintCmp{#3}{1}}% check to see if end has been reached
{}%
{\FactorTreeA\expandafter{\the\FactorTreeA
[{${#1}^{#2}$}]
%[1]%
}%
}%
{\FactorTreeA\expandafter{\the\FactorTreeA
[{${#1}^{#2}$}]
[{$#3$}%
}%
\FactorTreeB\expandafter{\the\FactorTreeB ]}%
}}%
}

\newcommand*{\FactorsToTree}[1]{\FactorsToTree@ #1}

% for factors displayed inline

\def\@factorinliner  #1{\@factorinliner@ #1}
\def\@factorinliner@ #1#2#3{\ifnum #2>1 \expandafter\@firstoftwo\else
\expandafter\@secondoftwo\fi
{{#1}^{#2}}{#1}}
\def\FactorsInline{%
\xintListWithSep\times
{\xintApply\@factorinliner{\expandafter\@gobble\factors}}%
}

\newlength{\horizontalshift}  % for positioning of the edges from the tree top
\newsavebox{\NandFactors}

\newcommand*{\FactorTree}[1]{%
\factorize{#1}%
\sbox{\NandFactors}{$#1=\FactorsInline$}%
\setlength{\horizontalshift}{(\wd\NandFactors-\widthof{$#1$})/2}%
\FactorTreeA{}%
\FactorTreeB{}%
\bracketset{action character=@}%
\xintApplyUnbraced\FactorsToTree{\expandafter\@gobble\factors}%
\begin{forest}
for tree={edge path={\noexpand\path [\forestoption{edge}]
(!u.south)--(.child anchor);}},
where={level()==1}
{edge path= {\noexpand\path [\forestoption{edge}]
($(!u.south)-(\the\horizontalshift,0cm)$)--(.child anchor);}}{},
where={n()==1}{draw,circle}{},
[{\box\NandFactors}, right=\horizontalshift,
@\the\FactorTreeA
@\the\FactorTreeB
]
\end{forest}
}

\makeatother

\begin{document}

\FactorTree{13}

\FactorTree{36}

\FactorTree{90}

\FactorTree{112}

\FactorTree{612}

\FactorTree{7875}

\FactorTree{22230}

\FactorTree{1073741824}

\FactorTree{2147483644}

\FactorTree{\xintiPow 2{40}}

\FactorTree{\xintNum{\xinttheexpr 2^{37}*3^{23}*17^{13}\relax}}%

% two examples ending with a (somewhat) big' prime,

\FactorTree{10968058712}

\FactorTree{1689242184972}

\end{document}

-
What is the point of \long\def\factorizeanddisplay#1{ instead of \def\factorizeanddisplay#1{? I.e., why use \long? [Answer to my own question: Never mind; I just found tex.stackexchange.com/questions/39450/… .] – Svend Tveskæg Sep 6 '13 at 5:00
@SvendTveskæg thanks and good point about \long : I just had a temporary brain collapse... will remove it! – jfbu Sep 6 '13 at 5:56
@PSTikZ you are right. I have now removed the circle. But I left the 1, partly for indecision regarding what to do when the last term is a pure prime power, partly because to have the last leaf as in your question it would be easier to construct \factors differently. – jfbu Sep 6 '13 at 13:04
In the first code of your post, is it necessary to load the packages fontenc and array? – Svend Tveskæg Sep 7 '13 at 4:32
@SvendTveskæg array is loaded to use the >{...} construct in the tabulars used in \displayfactorization. I think I loaded [T1]{fontenc} in the debugging phase as I was using \meaning\factors to check the contents of this macro, and it contains braces {, }, also there is a > in the output of \meaning. The factorization itself needs only package xint (and package xintexpr was loaded only to be able to use its \xinttheexpr....\relax construct, in order to test \factorize on randomly chosen products). – jfbu Sep 7 '13 at 6:16

A basic forest/count-based implementation.

Although forest provides keys to evaluate stuff and has basic if keys, I used a macro that evaluates everything. As it relies on counts, I don’t think it works with forest’s keys as they use pgfmath. (Which suffers from the typical problem that it uses TeX’s dimen registers which cannot be greater than 18-thousand-something; with the fp package and TikZ’s fixedpointarithmetic library this could be extended.)

The greatest representable number is 2 147 483 644 (2^31-4), the next integer 2 147 483 645 (2^31-3) fails.

The algorithm is not very efficient.
For the input number p, the factors 2, 3, 5, 7, 9, …, 2n + 1 until p/2 are checked whether they divide p without remainder.

The code includes some comments on the algorithm.

The \num macro from the siunitx package was used to typeset the numbers (even the exponents even though they cannot be greater than 30).

## Code

\documentclass[tikz]{standalone}
\usepackage{forest,mathtools,siunitx}
\makeatletter
\def\ifNum#1{\ifnum#1\relax
\expandafter\pgfutil@firstoftwo\else
\expandafter\pgfutil@secondoftwo\fi}
\forestset{
num content/.style={
delay={
content/.expanded={\noexpand\num{\forestoption{content}}}}},
pt@prime/.style={draw, circle},
pt@start/.style={},
pt@normal/.style={},
start primeTree/.style={%
/utils/exec=%
% \pt@start holds the current minimum factor, we'll start with 2
\def\pt@start{2}%
% \pt@result will hold the to-be-typeset factorization, we'll start with
% \pgfutil@gobble since we don't want a initial \times
\let\pt@result\pgfutil@gobble
% \pt@start@cnt holds the number of ^factors for the current factor
\def\pt@start@cnt{0}%
% \pt@lStart will later hold "l"ast factor used
\let\pt@lStart\pgfutil@empty,
alias=pt-start,
pt@start/.try,
delay={content/.expanded={$\noexpand\num{\forestove{content}} \noexpand\mathrlap{{}= \noexpand\pt@result}$}},
primeTree},
primeTree/.code=%
% take the content of the node and save it in the count
\c@pgf@counta\forestove{content}\relax
% if it's 2 we're already finished with the factorization
\ifNum{\c@pgf@counta=2}{%
% add the factor
\pt@addfactor{2}%
% finalize the factorization of the result
\pt@addfactor{}%
% and set the style to the prime style
\forestset{pt@prime/.try}%
}{%
% this simply calculates content/2 and saves it in \pt@end
% this is later used for an early break of the recursion since no factor
% can be greater then content/2 (for integers of course)
\edef\pt@content{\the\c@pgf@counta}%
\divide\c@pgf@counta2\relax
\advance\c@pgf@counta1\relax % to be on the safe side
\edef\pt@end{\the\c@pgf@counta}%
\pt@do}}

%%% our main "function"
\def\pt@do{%
% let's test if the current factor is already greather then the max factor
\ifNum{\pt@end<\pt@start}{%
% great, we're finished, the same as above
\expandafter\pt@addfactor\expandafter{\pt@content}%
\pt@addfactor{}%
\def\pt@next{\forestset{pt@prime/.try}}%
}{%
% this calculates int(content/factor)*factor
% if factor is a factor of content (without remainder), the result will
% equal content. The int(content/factor) is saved in \pgf@temp.
\c@pgf@counta\pt@content\relax
\divide\c@pgf@counta\pt@start\relax
\edef\pgf@temp{\the\c@pgf@counta}%
\multiply\c@pgf@counta\pt@start\relax
\ifNum{\the\c@pgf@counta=\pt@content}{%
% yeah, we found a factor, add it to the result and ...
\expandafter\pt@addfactor\expandafter{\pt@start}%
% ... add the factor as the first child with style pt@prime
% and the result of int(content/factor) as another child.
\edef\pt@next{\noexpand\forestset{%
append={[\pt@start, pt@prime/.try]},
append={[\pgf@temp, pt@normal/.try]},
% forest is complex, this makes sure that for the second child, the
% primeTree style is not executed too early (there must be a better way).
delay={
for descendants={
delay={if n'=1{primeTree, num content}{}}}}}}%
}{%
% Alright this is not a factor, let's get the next factor
\ifNum{\pt@start=2}{%
% if the previous factor was 2, the next one will be 3
\def\pt@start{3}%
}{%
% hmm, the previos factor was not 2,
% let's add 2, maybe we'll hit the next prime number
% and maybe a factor
\c@pgf@counta\pt@start
\advance\c@pgf@counta2\relax
\edef\pt@start{\the\c@pgf@counta}%
}%
% let's do that again
\let\pt@next\pt@do
}%
}%
\pt@next
}

%%% this builds the \pt@result macro with the factors
\def\pt@addfactor#1{%
\def\pgf@tempa{#1}%
% is it the same factor as the previous one
\ifx\pgf@tempa\pt@lStart
% add 1 to the counter
\c@pgf@counta\pt@start@cnt\relax
\advance\c@pgf@counta1\relax
\edef\pt@start@cnt{\the\c@pgf@counta}%
\else
% a new factor! Add the previous one to the product of factors
\ifx\pt@lStart\pgfutil@empty\else
% as long as there actually is one, the \ifnum makes sure we do not add ^1
\edef\pgf@tempa{\noexpand\num{\pt@lStart}\ifnum\pt@start@cnt>1
^{\noexpand\num{\pt@start@cnt}}\fi}%
\expandafter\pt@addfactor@\expandafter{\pgf@tempa}%
\fi
% setup the macros for the next round
\def\pt@lStart{#1}% <- current (new) factor
\def\pt@start@cnt{1}% <- first time
\fi
}
%%% This simply appends "\times #1" to \pt@result, with etoolbox this would be
%%% \appto\pt@result{\times#1}
\def\pt@addfactor@#1{%
\expandafter\def\expandafter\pt@result\expandafter{\pt@result \times #1}}

%%% Our main macro:
%%% #1 = possible optional argument for forest (can be tikz too)
%%% #2 = the number to factorize
\newcommand*{\PrimeTree}[2][]{%
\begin{forest}%
% as the result is set via \mathrlap it doesn't update the bounding box
% let's fix this:
tikz={execute at end scope={\pgfmathparse{width("${}=\pt@result$")}%
\path ([xshift=\pgfmathresult pt]pt-start.east);}},
% other optional arguments
#1
% And go!
[#2, start primeTree]
\end{forest}}
\makeatother
\begin{document}
\PrimeTree{36}
\PrimeTree{90}
\PrimeTree{112}
\PrimeTree{612}
\PrimeTree{7875}
\PrimeTree{22230}
\PrimeTree{1073741824}
\PrimeTree{2147483644}
\end{document}


## Output

-
@Qrrbrbirlbel: Could you add some comments to the code, so it will be less cryptic? – Dror Sep 5 '13 at 14:21
@Qrrbrbirlbel: A bit of annotating the code could be of great help. Where the algorithm starts, where the styling of the nodes end etc. – Dror Sep 5 '13 at 15:01
@Dror See my update. – Qrrbrbirlbel Sep 5 '13 at 15:30
Really nice! If I want to 'cluster' the digits in both the input number and the factors by using the \num command from siunitx, where should I put \num in your splendid code? (I have tried a few places, more or less at random, without success.) – Svend Tveskæg Sep 5 '13 at 17:50
@SvendTveskæg Cluster? Either way, I have updated my answer with siunitx\num macro (best to simply search for it). Even though, siunitx can parse products I haven't used it as the exponent is also included in the product but which base changes every time. Is this what you wanted? – Qrrbrbirlbel Sep 5 '13 at 18:42

This is half the answer, it just factorizes numbers. TeX is Turing-complete, no extra packages are required. :-) I'm not really in the mood for pst-tree, so I'll skip the second half. Besides, @Qrrbrbirlbel's answer (which I upvoted) covers that handsomely. If, for some reason, one prefers my code, it should be easy to patch the generic factor processing macros that I have with whatever is needed to generate the trees.

The algorithm I used is a quite simple one: it checks 2 and 3 and then checks all numbers that are 6k-1 or 6k+1, up to the square root of n. It is supposed to work with arbitrarily long numbers (up to 2^31, TeX's limit for counters), provided it does not exhaust TeX's memory when recursing. This happens, e.g., when you give it 2^31-1, which happens to be a Mersenne prime.

\documentclass[12pt]{article}

\makeatletter

\newcount\factorize@n  % the number to be factorized
\newcount\factorize@t  % temporary
\newcount\factorize@p  % a candidate factor
\newcount\factorize@c  % counter of factors

% machinery for factorization
%
\def\factorize#1{%
\factorize@begin{#1}%
\factorize@n=#1
\factorize@c=0
\factorize@p=2
\factorize@try%
\factorize@p=3
\factorize@try%
\factorize@p=5
\factorize@loop%
\ifnum\factorize@c>0%
\ifnum\factorize@n>1%
\factorize@process{\the\factorize@n}%
\fi%
\else%
\factorize@process{\the\factorize@n}%
\fi%
\factorize@end{#1}%
}
\def\factorize@loop{%
\factorize@t=\factorize@p
\multiply\factorize@t by \factorize@p
\ifnum\factorize@t>\factorize@n\else%
\factorize@try%
\advance\factorize@p by 2
\factorize@t=\factorize@p
\multiply\factorize@t by \factorize@p
\ifnum\factorize@t>\factorize@n\else%
\factorize@try%
\advance\factorize@p by 4
\factorize@loop%
\fi%
\fi%
}

\def\factorize@try{%
\factorize@t=\factorize@n
\divide\factorize@t by \factorize@p
\multiply\factorize@t by \factorize@p
\ifnum\factorize@n=\factorize@t%
\factorize@process{\the\factorize@p}%
\divide\factorize@n by \factorize@p
\advance\factorize@c by 1
\factorize@try%
\fi%
}

% Stubs to be called at start, end, and when a factor is found
%
\def\factorize@begin#1{%
\noindent%
$#1 =%$
}
\def\factorize@end#1{%
$%$
\par%
}
\def\factorize@process#1{%
\ifnum\factorize@c>0%
\times%
\fi%
#1%
}

\makeatother

% Demo
%
\begin{document}
\factorize{42}
\factorize{5040}
\factorize{1234567890}
%\factorize{2147483647} exhausts TeX's memory
\end{document}


A small variation groups factors of multiplicity greater than one.

\documentclass[12pt]{article}

\makeatletter

\newcount\factorize@n  % the number to be factorized
\newcount\factorize@t  % temporary
\newcount\factorize@p  % a candidate factor
\newcount\factorize@c  % counter of factors (total)
\newcount\factorize@w  % counter of factors (power of given candidate)

% machinery for factorization
%
\def\factorize#1{%
\factorize@begin{#1}%
\factorize@n=#1
\factorize@c=0
\factorize@p=2
\factorize@try%
\factorize@p=3
\factorize@try%
\factorize@p=5
\factorize@loop%
\ifnum\factorize@c>0
\ifnum\factorize@n>1
\factorize@process{\the\factorize@n}{1}%
\fi%
\else%
\factorize@process{\the\factorize@n}{1}%
\fi%
\factorize@end{#1}%
}
\def\factorize@loop{%
\factorize@t=\factorize@p
\multiply\factorize@t by \factorize@p
\ifnum\factorize@t>\factorize@n\else%
\factorize@try%
\advance\factorize@p by 2
\factorize@t=\factorize@p
\multiply\factorize@t by \factorize@p
\ifnum\factorize@t>\factorize@n\else%
\factorize@try%
\advance\factorize@p by 4
\factorize@loop%
\fi%
\fi%
}
\def\factorize@try{%
\factorize@w=0
\factorize@try@aux%
\ifnum\factorize@w>0
\factorize@process{\the\factorize@p}{\the\factorize@w}%
\advance\factorize@c by \factorize@w
\fi%
}
\def\factorize@try@aux{%
\factorize@t=\factorize@n
\divide\factorize@t by \factorize@p
\multiply\factorize@t by \factorize@p
\ifnum\factorize@n=\factorize@t
\divide\factorize@n by \factorize@p
\advance\factorize@w by 1
\factorize@try@aux%
\fi%
}

% Stubs to be called at start, end, and when a factor is found
%
\def\factorize@begin#1{%
\noindent%
$#1 =%$
}
\def\factorize@end#1{%
$%$
\par%
}
\def\factorize@process#1#2{%
\ifnum\factorize@c>0
\times%
\fi%
\ifnum#2>1
#1^{#2}%
\else%
#1%
\fi%
}

\makeatother

% Demo
%
\begin{document}
\factorize{42}
\factorize{5040}
\factorize{1234567890}
%\factorize{2147483647} exhausts TeX's memory
\end{document}

-
This is nice! Is it possible to get similar factors as powers, say, 8 = 2^3 instead of 8 = 2 \times 2 \times 2? – Svend Tveskæg Sep 5 '13 at 17:39
Yes, you would simply use a counter in \factorize@try and call \factorize@process there only once, when finishing. – nickie Sep 5 '13 at 17:53
I'm not good at coding in LaTeX at all; can I make you either edit your answer or create another one with this feature? :) – Svend Tveskæg Sep 5 '13 at 17:56
OK, added that too. – nickie Sep 5 '13 at 20:18
Perfect! (I really like the fact that no packages are loaded.) – Svend Tveskæg Sep 5 '13 at 21:06

MWE with Asymptote solution (not optimized). struct PrimeTree that draws a tree is defined in the preamble and is used in asy pictures. It consists of a function fillPrimeList(), which fills the list of prime numbers, a function factors(), which collects prime factors of the number, and a function draw(), which draws a tree.

% pfactors.tex :
\documentclass{article}
\usepackage{subcaption}
\usepackage{lmodern}
\usepackage[inline]{asymptote}
\usepackage[left=2cm,right=2cm]{geometry}
\begin{asydef}
struct PrimeTree{
int num;
int maxPrimeInd;
int[] primelist;
pen defpen,linkPen;

void fillPrimeList(){
int numd=num;
int nmax;
bool primetest;
if(numd%2==0)numd=(int)(numd/2);
if(numd%3==0)numd=(int)(numd/3);
nmax=(int)(floor(sqrt(numd)));
primelist=new int[nmax];
primelist[0]=2;
primelist[1]=3;

maxPrimeInd=2;

for(int i=5;i<=nmax;i+=2){
primetest=true;
for(int j=0;primetest && (j<maxPrimeInd);++j){
if((i%primelist[j]==0)){
primetest=false;
}
}
if(primetest){
if(numd%i==0){
numd=(int)(numd/i);
nmax=(int)(floor(sqrt(numd)));
}
primelist[maxPrimeInd]=i;
++maxPrimeInd;
}
}
}

int[] factors(){
fillPrimeList();
int[] fs;
int a,b;
a=num;
for(int i=0;i<maxPrimeInd;++i){
b=primelist[i];
while(a%b==0){
fs.push(b);
a=(int)(a/b);
}
}
if(a>1)fs.push(a);
return fs;
}

void draw(){
int[] fs=factors();
int a,b;

real row=0, col=0;
real dr=1.618, dc=1.382;

a=num;
for(int i=0; a>1 && i<fs.length;++i){
b=fs[i];
if(a!=b){
label(string(a),(0,row),W);
draw(Label(string(b),(col+dc,row),E),roundbox);
draw((0,row)--(col+dc,row),linkPen);
if(i>0) draw((0,row)--(0,row+dr),linkPen);
}else{
draw(Label(string(b),(col+dc,row),E),roundbox);
if(i>0) draw((0,row+dr)--(col+dc,row),linkPen);
}
row-=dr;
a=(int)(a/b);
}
}
void operator init(int num, pen defpen=darkblue+fontsize(10pt), pen linkPen=orange){
assert(num>0);
this.num=num;
this.defpen=defpen;
this.linkPen=linkPen;
defaultpen(defpen);
draw();
}
}

\end{asydef}

\begin{document}
\begin{figure}
\captionsetup[subfigure]{justification=centering}
\centering
\begin{subfigure}{0.24\textwidth}
\begin{asy}
unitsize(10pt);
PrimeTree(22230);
\end{asy}
\caption{}
\label{fig:1a}
\end{subfigure}
%
\begin{subfigure}{0.24\textwidth}
\begin{asy}
unitsize(10pt);
PrimeTree(1073741824);
\end{asy}
\caption{}
\label{fig:1b}
\end{subfigure}
%
\begin{subfigure}{0.24\textwidth}
\begin{asy}
unitsize(10pt);
PrimeTree(2147483644);
\end{asy}
\caption{}
\label{fig:1c}
\end{subfigure}
%
\begin{subfigure}{0.24\textwidth}
\begin{asy}
unitsize(10pt);
PrimeTree(19928968819);
\end{asy}
\caption{}
\label{fig:1d}
\end{subfigure}
\end{figure}
\end{document}
%
% To process it with latexmk, create file latexmkrc:
%

else: % avoiding vertical overlapping with box above
pair endpos;
numeric r; r = radius(textext(str_number(n)));
endpos = pos - (0, voffset - v + max(r1,r));
draw pos--endpos;
circledlabel(str_number n, endpos);
fi
enddef;

def prime_tree(expr n) =
numeric v, h, cnt, N;
h = 0.75cm; v = -.65cm; cnt = 0; N = n;
prime_factorization(2, n);
enddef;

beginfig(1);
numeric L; L = 4cm; numeric l; l := -5cm;
pair pos;
pos := origin; prime_tree(36);
pos := (L, 0); prime_tree(90);
pos := (2L, 0); prime_tree(112);
pos := (3L, 0); prime_tree(612);
L := 4.5cm;
pos := (0, l); prime_tree(7875);
pos := (L, l); prime_tree(22230);
pos := (2L, l); prime_tree(765434567654565);
endfig;

\end{mplibcode}
\end{document}


Output:

-