The method of computation of

[1] Maximal product of primes whose sum is bounded.See also the paper written by

[2] Landau's function for one million billions.

Let

If

There are 5 types for the computation of

Type 0 : ifn'= por_{k+1}-2n'= pthen_{k+1}-1G(p._{k},n')= p_{k+1}/2

Type 1 : if0 ≤ n' < pthen_{k+1}- p_{k}G(p._{k},n')=1

Type 2 : ifpis a prime, say_{k+1}-mmq, thenG(p._{k},n')=p_{k+1}/q

Type 3 : is the general case, solved by the δ-method (see [1] or [2] for explanations).

Type 4 : is the case where the δ-method fails and the combinatorial method is used. This case occurs only whenn'is small.

The variableecartis the largest integer such thath(n-ecart) =h(n).

All the prime factors ofG(p(but one) are close to_{k},n')p, i.e. are equal to_{k}pwith_{k+i}ismall. The values of thei's are given in a list. For instance the display

[1,4], [ p',p''], [-2], [p''',q] means thatG(pwith_{k}, n')=p' p'' / p''' qp'=p,_{k+1}p''=pand_{k+4}p'=p._{k-2}

To use the Maple program to compute

Open a MAPLE sheet and type read calculhden:

♣ If you want to calculateh(n)for a not too largen, sayn≤10, then type hdennext(n);^{12}

The procedurehdennextgenerates the primes by using the functionnextprime.

♦ If you want to calculateh(n)for several not too largen's, you had better to generate the primes and the sum of consecutive primes with a sieve. For that, for, say,nmax=10, type crible(nmax);^{7}

Then, for several values ofn, type hdensieve(n);

♥ If you want to calculateh(2for^{a})amin≤a≤amaxwith1≤amin ≤ amax ≤ 117, then type hdedeuxpuisa(amin,amax);

This procedure has been used to write the above mentionned filepowersof2and10. For1≤a ≤ 117, andn=2, the values of^{a}pand_{k}n'have been precomputed and written in the filecalculhden, see the reference [1].

♠ If you want to calculateh(10for^{a})amin≤a≤amaxwith1≤amin ≤ amax ≤ 35, then type hdedixpuisa(amin,amax);

This procedure has been used to write the above mentionned filepowersof2and10. For1≤a ≤ 35, andn=10, the values of^{a}pand_{k}n'have been precomputed and written in the filecalculhden, see the reference [1].

¶ If you want to calculateG(p,m), then type Gpkmh(p,m,500,2);.

You may try to change the two last paramaters; the last one is a printing parameter : increasing it will print more partial results.