Step by Step solution about How to resolve the algorithm Proper divisors step by step in the Mathematica / Wolfram Language programming language

Code Explanation:

ProperDivisors[n] Function:

• This function takes an integer n as input and calculates the proper divisors of n.
• Proper divisors are all positive divisors of n except n itself.
• The function returns a list of these divisors using Most@Divisors@n.

Grid[Table[{n, ProperDivisors[n]}]]:

• This code creates a tabular representation of n and its proper divisors.
• It generates a table with two columns: n and ProperDivisors[n]
• The Table function ranges over n values from 1 to 10.

Fold[Last[SortBy[{#1, {#2, Length@ProperDivisors[#2]}}, Last]] &, {0, 0}, Range[20000]]

• This code finds the integer n with the most number of proper divisors in the range from 1 to 20000.
• Fold applies a function repeatedly to a list. In this case, it applies Last[SortBy[...]] to the list {0, 0} and the elements of Range[20000].
• Last[SortBy[...]] sorts the list by the second element of each pair, which is the number of proper divisors for each n. It then returns the last element, which is the pair with the maximum number of divisors.

Last@SortBy[Table[{n, Length@ProperDivisors[n]}, {n, 1, 20000}], Last]

• This alternative method also finds the integer n with the most proper divisors in the range 1 to 20000.
• Table generates a list of pairs {n, Length@ProperDivisors[n]} for each n in the range.
• SortBy sorts the list by the second element (number of divisors) in descending order.
• Last extracts the last element of the sorted list, which has the maximum number of divisors.

ProperDivisors[n_Integer /; n > 0] := Most@Divisors@n;


Grid@Table[{n, ProperDivisors[n]}, {n, 1, 10}]


Fold[
 Last[SortBy[{#1, {#2, Length@ProperDivisors[#2]}}, Last]] &,
 {0, 0},
 Range[20000]]


Last@SortBy[
  Table[
    {n, Length@ProperDivisors[n]},
    {n, 1, 20000}],
  Last]