The following iterative sequence is defined for the set of positive integers:

n → n/2 (n is even)

n → 3n + 1 (n is odd)

Using the rule above and starting with 13, we generate the following sequence:
13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.

– Project Euler Problem 14

function getChain(n::UInt64)
        chain = Array{UInt64,1}();
        push!(chain, n);
        while n > 1
                if n%2 == 0
                        n /= 2;
                else
                        n = 3*n+1;
                end
                push!(chain,n);
        end
        return chain
end


function getLongestChain(n_max::UInt64)
        longestChainLength::UInt64 = 0;
        longestChain = Array{UInt64,1}()
        for n in 1:1:n_max
                chain = getChain(n);
                if length(chain) > longestChainLength
                        longestChain = chain;
                        longestChainLength = length(longestChain);
                end
        end
        return longestChain, longestChainLength
end

function parseARGS()
        if isempty(ARGS)
                n_max::UInt64 = 1000000;
        else
                n_max = tryparse(UInt64,ARGS[1]);
        end
        return n_max
end

function main()
        n_max = parseARGS();
        longestChain, longestChainLength = getLongestChain(n_max);
        println("longest chain starting number: ",longestChain[1]);
        println("length: ", longestChainLength);
        println("longest chain:");
        for element in longestChain[1:end-1]
                print(string(element), "->")
        end
        print(string(longestChain[end]));
end


main()

$ julia ProjectEuler0014.jl 13
longest chain starting number: 9
length: 20
longest chain: 
9->28->14->7->22->11->34->17->52->26->13->40->20->10->5->16->8->4->2->1

0.000322 seconds (158 allocations: 8.188 KiB)

$ julia ProjectEuler0014.jl 1000000
longest chain starting number: 837799
length: 525
longest chain: 
837799->2513398->1256699->3770098->1885049->5655148->2827574->1413787->4241362->2120681->6362044->3181022->1590511->4771534->2385767->7157302->3578651->10735954->5367977->16103932->8051966->4025983->12077950->6038975->18116926->9058463->27175390->13587695->40763086->20381543->61144630->30572315->91716946->45858473->137575420->68787710->34393855->103181566->51590783->154772350->77386175->232158526->116079263->348237790->174118895->522356686->261178343->783535030->391767515->1175302546->587651273->1762953820->881476910->440738455->1322215366->661107683->1983323050->991661525->2974984576->1487492288->743746144->371873072->185936536->92968268->46484134->23242067->69726202->34863101->104589304->52294652->26147326->13073663->39220990->19610495->58831486->29415743->88247230->44123615->132370846->66185423->198556270->99278135->297834406->148917203->446751610->223375805->670127416->335063708->167531854->83765927->251297782->125648891->376946674->188473337->565420012->282710006->141355003->424065010->212032505->636097516->318048758->159024379->477073138->238536569->715609708->357804854->178902427->536707282->268353641->805060924->402530462->201265231->603795694->301897847->905693542->452846771->1358540314->679270157->2037810472->1018905236->509452618->254726309->764178928->382089464->191044732->95522366->47761183->143283550->71641775->214925326->107462663->322387990->161193995->483581986->241790993->725372980->362686490->181343245->544029736->272014868->136007434->68003717->204011152->102005576->51002788->25501394->12750697->38252092->19126046->9563023->28689070->14344535->43033606->21516803->64550410->32275205->96825616->48412808->24206404->12103202->6051601->18154804->9077402->4538701->13616104->6808052->3404026->1702013->5106040->2553020->1276510->638255->1914766->957383->2872150->1436075->4308226->2154113->6462340->3231170->1615585->4846756->2423378->1211689->3635068->1817534->908767->2726302->1363151->4089454->2044727->6134182->3067091->9201274->4600637->13801912->6900956->3450478->1725239->5175718->2587859->7763578->3881789->11645368->5822684->2911342->1455671->4367014->2183507->6550522->3275261->9825784->4912892->2456446->1228223->3684670->1842335->5527006->2763503->8290510->4145255->12435766->6217883->18653650->9326825->27980476->13990238->6995119->20985358->10492679->31478038->15739019->47217058->23608529->70825588->35412794->17706397->53119192->26559596->13279798->6639899->19919698->9959849->29879548->14939774->7469887->22409662->11204831->33614494->16807247->50421742->25210871->75632614->37816307->113448922->56724461->170173384->85086692->42543346->21271673->63815020->31907510->15953755->47861266->23930633->71791900->35895950->17947975->53843926->26921963->80765890->40382945->121148836->60574418->30287209->90861628->45430814->22715407->68146222->34073111->102219334->51109667->153329002->76664501->229993504->114996752->57498376->28749188->14374594->7187297->21561892->10780946->5390473->16171420->8085710->4042855->12128566->6064283->18192850->9096425->27289276->13644638->6822319->20466958->10233479->30700438->15350219->46050658->23025329->69075988->34537994->17268997->51806992->25903496->12951748->6475874->3237937->9713812->4856906->2428453->7285360->3642680->1821340->910670->455335->1366006->683003->2049010->1024505->3073516->1536758->768379->2305138->1152569->3457708->1728854->864427->2593282->1296641->3889924->1944962->972481->2917444->1458722->729361->2188084->1094042->547021->1641064->820532->410266->205133->615400->307700->153850->76925->230776->115388->57694->28847->86542->43271->129814->64907->194722->97361->292084->146042->73021->219064->109532->54766->27383->82150->41075->123226->61613->184840->92420->46210->23105->69316->34658->17329->51988->25994->12997->38992->19496->9748->4874->2437->7312->3656->1828->914->457->1372->686->343->1030->515->1546->773->2320->1160->580->290->145->436->218->109->328->164->82->41->124->62->31->94->47->142->71->214->107->322->161->484->242->121->364->182->91->274->137->412->206->103->310->155->466->233->700->350->175->526->263->790->395->1186->593->1780->890->445->1336->668->334->167->502->251->754->377->1132->566->283->850->425->1276->638->319->958->479->1438->719->2158->1079->3238->1619->4858->2429->7288->3644->1822->911->2734->1367->4102->2051->6154->3077->9232->4616->2308->1154->577->1732->866->433->1300->650->325->976->488->244->122->61->184->92->46->23->70->35->106->53->160->80->40->20->10->5->16->8->4->2->1

6.126916 seconds (7.41 M allocations: 3.104 GiB, 1.67% gc time)

Another recreational math exercise without any application? If you can think of one, please let me know ;).