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performance tuning - Happy 2K prime question


This being the Q number 2K in the site, and this being the day we got the confirmation of mathematica.se graduating soon, I think a celebration question is in order.


So...



What is the fastest way to compute the happy prime number 2000 in Mathematica?


Edit


Here are the Timing[ ] results so far:


 {"JM",      {5.610, 137653}}
{"Leonid", {5.109, {12814, 137653}}}
{"wxffles", {4.11, {12814, 137653}}}
{"Rojo", {0.765, {12814, 137653}}}
{"Rojo1", {0.547, {12814, 137653}}}

Answer



This answer should be read upside down, since the last edit has the fastest, neatest and shortest answer



Module[{$guard = True},

happyQ[i_] /; $guard := Block[{$guard = False, appeared},
appeared[_] = False;
happyQ[i]
]
]

e : happyQ[_?appeared] := e = False;


happyQ[1] = True;

e : happyQ[i_] := e = (appeared[i] = True; happyQ[#.#&@IntegerDigits[i]])

Now, taking this from @LeonidShiffrin


happyPrimeN[n_] := Module[{m = 0, pctr = 0},
While[m < n, If[happyQ@Prime[++pctr], m++]];
{pctr, Prime[pctr]}];

EDIT



Ok, this was cool, but if you don't mind wasting a little memory and not resetting appeared, it becomes simple and less cool


appeared[_] = False;
happyQ[1] = True;
happyQ[_?appeared] = False;
e : happyQ[i_] := e = (appeared[i] = True; happyQ[#.# &@IntegerDigits[i]])

EDIT2


Slightly faster but I like it twice as much


happyQ[1] = True;
e : happyQ[i_] := (e = False; e = happyQ[#.# &@IntegerDigits[i]])


or perhaps to make it slightly shorter and a little bit more memory efficient, reducing the recursion tree's height


happyQ[1] = True;
e : happyQ[i_] := e = happyQ[e = False; #.# &@IntegerDigits[i]]

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