# Show Ulam's spiral

Similar to this but in this one you need to write spiral starting from the center where:

• space means composite number
• .(dot) means prime number.

Size of the spiral can be given as parameter or on stdin. By size I mean side of square NxN and also N will be always odd.

• Doesn't Ulam's spiral usually start with 1 in the middle and spiral outward, instead of spiraling inward? Commented Jan 8, 2012 at 21:32
• Yeah, it's late so my brain is out of order. Commented Jan 8, 2012 at 22:17
• By size of the spiral do you mean the number to go up to, or the number of spirals? Commented Jan 8, 2012 at 22:24
• Also, are there limits on the number that may be input or is it 1 to infinity? Commented Jan 8, 2012 at 22:29
• Square NxN and limit to 100000 Commented Jan 8, 2012 at 22:50

## Python, 219 chars

N=input()
A={0:' '}
d=1j
x=1
for p in range(2,N*N+1):
A[x]=' .'[all(p%i for i in range(2,p))]
if abs(x.imag)==abs(x.real):x+=(1-1j)*(d==1);d*=1j
x+=d
R=range(N)
for y in R:print''.join(A[x-N/2+(N/2-y)*1j]for x in R)


Works for any odd N. For example:

$echo 9 | ./ulam.py . . . . . . . . . . . .. . . . . . . . . .  JavaScript (240 202 195 151 characters) Update: Another much smaller version without function (a lot of credits to @mellamokb): for(x=3,e=d=f=a>>1,c=2;(x&1?x&2?++e<a-d:--e>d:x&2?++f<a-d-1:--f>d)||++x&3||d--;c ++)for(g=0;g<2*a*a;z[g+=c]=1)z[c]||z.getContext("2d").fillRect(e,f,1,1)  Works with this HTML: <script>a = 50</script> <canvas id=z width=50 height=50></canvas>  25x25 example (zoomed in) - 800x800 example This new version now performs well and outputs the right size (NxN) for any odd a. Found some small improvements (195 now). Thanks @mellamokb. Old version: c=1;i=e=0;b={};for(d=[];c<a*a;){d.push("");for(i+=e+=2;i--;)d[Math.min(e-2,i)]+= j();d.unshift("");for(i-=e;++i<e;)d[g=Math.max(0,i)]=j()+d[g]}x.innerHTML=d.join ("\n");function j(){if(f=!b[++c])for(h=c*c;h<2*a*a;h+=c)b[h]=1;return f?".":" "}  Currently takes variable a as input and outputs to an element with the id x: <script>a = 50</script> <pre id=x>  I used the Sieve of Eratosthenes for prime generation, which works really well. Output is quite slow so far though. Don't expect this to run for huge n yet. • I've used Math.max(0,i) trick in the past before thinking it was clever, but it's actually shorter to use a ternary: d[g=i>0?i:0]. Same with Math.min(e-2,i) which should be rewritten as d[i<e-2?i:e-2]. Commented Jan 9, 2012 at 21:30 • @mellamokb: My never version doesn't use min or max anymore. But thanks for the tip; it's good to remember these minor optimizations. – copy Commented Jan 9, 2012 at 21:43 • This is code-golf, so sacrifice a little performance to gain 3 chars: for(g=0;g<2*a*a;b[g+=c]=1); Likewise, move the whole for outside of the if and you can nix the {} for the if, saving another 2 characters :) Commented Jan 9, 2012 at 21:48 • @mellamokb: wow, you're good at this. The second suggestion was even better because it can now be written as b[c]||z.getContext, saving 2 more characters. – copy Commented Jan 9, 2012 at 22:21 • Working through some ideas. I have a few improvements to make, but I figured out this eval(["--f>d","--e>d","++f<a-d-1","++e<a-d"][x%4])||++x%4||d--;. Basically this replaces all the for(h().. and also changes the condition on the main loop to just d. Then you must define x=0 and subtract 1 from d to begin with since it's no longer in the condition. Finally, inline h() since it's now only called once, and should be able to save about 10 characters (hopefully) :). Will update with another idea that might save much more in a bit... Commented Jan 10, 2012 at 0:09 ## Golfscript - 92 Characters Based on my answer here: ~.(:S+,:R{S\-:|;R{S-:$|>' .'1/[|$.|]2/@:d|~)$<!^=~:$;:y.*4*$-y-)2d*$y-*+:$,{)\$\%!},,2==}%n}%


## APL (85)

K[R↑+\(1+M-⍨N×M←⌈N÷2),(2/⍳N)/(2×N)⍴1(-N)¯1N]←K←⍳R←N×N←⎕⋄'. '[1+N N⍴K∊P/⍨P∊P∘.×P←1↓⍳R]


Explanation:

• Generating the spiral:
• K←⍳R←N×N←⎕: Read N from the user. The array size N×N is stored in R. K is [1..R].
• (1+M-⍨N×M←⌈N÷2): The coordinate of the middle field.
• (2×N)1(-N)¯1N: the delta coordinates for the next field (i.e. 1 right, up a line (so N fields to the left in a 1-dimensional array), then 1 left, then down a line.
• (2/⍳N)/: duplicate the deltas to form an expanding spiral. 2/⍳N is 1 1 2 2 3 3 ... N N, duplicating the deltas by these values gives right up left left down down right right right...
• R↑+\: sum these values (giving absolute coordinates) and take the first R.
• K[...]←K: assign K to K in the order given above. We now have K in spiral order.
• Generating the pattern:
• P/⍨P∊P∘.×P←1↓⍳R: more or less the standard APL idiom for generating primes. P is [2..R], P∘.×P is a multiplication table for P. P∘.P therefore contains all composite numbers in the range [1..R]. P/⍨ then selects from P all values present in P∘.×P, giving a list of composite numbers.
• 1+N N⍴K∊: this selects from K all the composite numbers, giving a binary list in spiral order where there's an 1 if the number is composite. Then add 1 so that composite numbers are 2 and noncomposite (prime) numbers are 1. This is formatted as a N by N table.
• '. '[...]: prime numbers (1) become '.' and composites (2) become ' '.

# MATL, 13 bytes

UGlYLZp42*cwx


Try it online!

A clockwise outward spiral which starts from the right.

## Explanation

    U       % convert char matrix to numeric or to general array / square
G       % paste from user-input clipboard G
l       % array of ones
YL      % Create spiral matrix
Zp      % is prime / totient function
42      % number literal
*       % array product (element-wise, singleton expansion)
c       % convert to character array
w       % swap elements in stack
x       % delete elements
% (implicit) convert to string and display


## Python - 203 Characters

x=input();y=x-1;w=x+y