Are there any algorithms or methods to compute Landau function $g(n)$?

Question

The details about Landau function can be found in A000793.

Maybe there is some methods in A000793, but I don't understand what it says. If possible, can someone illustrate the method?

Raymond Manzoni · Accepted Answer · 2013-07-10T22:39:40.413

2

I considered Michael Somos' pari script and added some comments :

{  a(n) = 
   local(m, t, j, u); 
   if (n < 2, 
       n >= 0, \\the result 0 or 1
   \\else
       m = ceil( n/exp(1) ); 
       t = ceil( (n/m)^m ); 
       j=1;                  \\init of result
       for (i=2, t, 
           u = factor(i);    \\returns [p1,a1],[p2,a2],...,[pm,am] if i=p1^a1 p2^a2...pm^am
           u = sum( k=1, matsize(u)[1], u[k, 1]^u[k, 2]); \\sum k=1 to m of pk^ak
           if (u <= n, j=i)  \\if the sum is not larger than n memorize i
       ); 
       j   \\the result
   )
}

Note that pari/gp is free and rather convenient for tests !

This requires quite some factorizations. The referenced paper by Deleglise, Jean-Louis Nicolas and Paul Zimmermann 'Landau's function for one million billions' should be more interesting for large values.

edited Jul 10 '13 at 22:39

answered Jul 09 '13 at 15:17

Raymond Manzoni

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Pari/GP rather uses \\ for comments. – ccorn Jul 09 '13 at 16:34
Thanks to notice @ccorn. – Raymond Manzoni Jul 09 '13 at 18:04
The referenced paper gives hints with regard to the correctness of Michael Somos's approach. Thanks for that link. Upvoted. – ccorn Jul 10 '13 at 20:49
Thanks again @ccorn ! Let's add that Jean-Louis Nicolas proposes Maple code for $g(n)$'s evaluation at his homepage that could be useful too. – Raymond Manzoni Jul 10 '13 at 22:47
Glad it helped @JiabinHe! – Raymond Manzoni Jul 12 '13 at 14:25

Are there any algorithms or methods to compute Landau function $g(n)$?

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