I implemented a solution to the problem below in Mathematica, but it takes a very long time (hours) to compute f of kis or the set B for large numbers.
Somebody suggested that implementing this in C++ resulted in a solution in less than 10 minutes. Would C++ be a good language to learn to solve these problems, or can my Mathematica code be improved to fix the performance issues?
I don't know anything about C or C++ and it should be difficult to start to learn this languages. I prefer to improve or write new code in mathematica.
Problem Description
Let $f$ be an arithmetic function and
A={k1,k2,...,kn} are integers in
increasing order.
Now I want to start with k1 and
compare f(ki) with f(k1). If
f(ki)f(k1), put ki as k1.
Now start with ki, and compare f(kj)
with f(ki), for ji. If f(kj)f(ki),
put kj as ki, and repeat this
procedure.
At the end we will have a sub sequence
B={L1,...,Lm} of A by this property:
f(L(i+1))f(L(i)), for any 1<=i<=m-1
For example, let f is the divisor
function of integers.
Here I put some part of my code and this is just a sample and the question in my program could be more larger than these:
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f[n_] := DivisorSigma[0, n];
g[n_] := Product[Prime[i], {i, 1, PrimePi[n]}];
k1 = g[67757] g[353] g[59] g[19] g[11] g[7] g[5]^2 6^3 2^7;
k2 = g[67757] g[353] g[59] g[19] g[11] g[7] g[5] 6^5 2^7;
k3 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5] 6^4 2^7;
k4 = g[67759] g[349] g[53] g[19] g[11] g[7] g[5] 6^5 2^6;
k5 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5] 6^4 2^8;
k6 = g[67759] g[349] g[53] g[19] g[11] g[7] g[5]^2 6^3 2^7;
k7 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5] 6^5 2^6;
k8 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5] 6^4 2^9;
k9 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5]^2 6^3 2^7;
k10 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5] 6^5 2^7;
k11 = g[67759] g[349] g[53] g[19] g[11] g[7] g[5]^2 6^4 2^6;
k12 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5]^2 6^3 2^8;
k13 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5]^2 6^4 2^6;
k14 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5]^2 6^3 2^9;
k15 = g[67757] g[359] g[53] g[19] g[11] g[7] g[5]^2 6^4 2^7;
k16 = g[67757] g[359] g[53] g[23] g[11] g[7] g[5] 6^4 2^8;
k17 = g[67757] g[359] g[59] g[19] g[11] g[7] g[5] 6^4 2^7;
k18 = g[67757] g[359] g[53] g[23] g[11] g[7] g[5] 6^4 2^9;
k19 = g[67759] g[353] g[53] g[19] g[11] g[7] g[5] 6^4 2^6;
k20 = g[67763] g[347] g[53] g[19] g[11] g[7] g[5] 6^4 2^7;
k = Table[k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15, k16, k17, k18, k19, k20];
i = 1;
count = 0;
For[j = i, j <= 20, j++,
If[f[k[[j]]] - f[k[[i]]] > 0, i = j; Print["k",i];
count = count + 1]];
Print["count= ", count]
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