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

The provided Wolfram code is used to generate Zumkeller numbers. Zumkeller numbers are a sequence of positive integers for which the sum of the divisors of the number is an odd number and the central coefficient of the divisor polynomial is positive.

The code begins by defining a function called ZumkellerQ that takes a positive integer n as input and returns True if n is a Zumkeller number and False otherwise.

Inside the ZumkellerQ function:

1. It calculates the divisors of n and stores them in the list d.
2. It calculates the sum of the divisors and stores it in the variable ds.
3. It checks if ds is odd. If it is odd, the function returns False because a Zumkeller number must have an odd sum of divisors.
4. If ds is even, it constructs a polynomial using the divisors in d and stores the coefficients of the polynomial in the list t.
5. It checks if the central coefficient of the polynomial t[[1 + ds/2]] is positive. If it is positive, the function returns True, indicating that n is a Zumkeller number.

After defining the ZumkellerQ function, the code generates two lists of Zumkeller numbers:

1. The first list, res, contains the first 220 Zumkeller numbers. It does this by iterating through positive integers, checking if each integer is a Zumkeller number using the ZumkellerQ function, and adding it to the list if it is.
2. The second list, res, contains the first 40 odd Zumkeller numbers. It does this by iterating through odd positive integers, checking if each integer is a Zumkeller number, and adding it to the list if it is.

Finally, the code prints the two lists of Zumkeller numbers.

ClearAll[ZumkellerQ]
ZumkellerQ[n_] := Module[{d = Divisors[n], t, ds, x},
   ds = Total[d];
   If[Mod[ds, 2] == 1,
    False
    ,
    t = CoefficientList[Product[1 + x^i, {i, d}], x];
    t[[1 + ds/2]] > 0
    ]
   ];
i = 1;
res = {};
While[Length[res] < 220,
  r = ZumkellerQ[i];
  If[r, AppendTo[res, i]];
  i++;
  ];
res

i = 1;
res = {};
While[Length[res] < 40,
  r = ZumkellerQ[i];
  If[r, AppendTo[res, i]];
  i += 2;
  ];
res