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

This code defines a function called DuffianQ that takes an integer n as input and checks if it is a Duffian number. A Duffian number is a composite (non-prime) number such that the sum of its divisors is coprime with the number itself.

The code first clears any previously defined symbols with the name DuffianQ using ClearAll[DuffianQ].

The function DuffianQ is then defined as follows:

DuffianQ[n_Integer] := CompositeQ[n] \[And] CoprimeQ[DivisorSigma[1, n], n]

Here, CompositeQ[n] checks if n is a composite number (not prime), and CoprimeQ[DivisorSigma[1, n], n] checks if the sum of the divisors of n (excluding 1 and n itself) is coprime with n.

Next, the code uses the function Select[DuffianQ][Range[1000000]] to find all Duffian numbers up to 1 million. The result is stored in the variable dns.

The code then takes the first 50 Duffian numbers from the list dns using Take[dns, UpTo[50]].

The code then defines a function called triplets that takes a list of Duffian numbers as input and returns a list of strings, where each string consists of two Duffian numbers separated by a dash. The function uses the ToString function to convert the Duffian numbers to strings, and the \[LongDash] symbol to represent the dash.

The code then uses the SequencePosition function to find all occurrences of the pattern {1, 1} in the list of differences between consecutive Duffian numbers in dns. The result is a list of positions, which is then used to extract the corresponding triplets of Duffian numbers.

Finally, the code uses the Multicolumn function to display the list of triplets in a table with 5 columns and an automatic width for each column. The Appearance -> "Horizontal" option is used to display the table horizontally.

ClearAll[DuffianQ]
DuffianQ[n_Integer] := CompositeQ[n] \[And] CoprimeQ[DivisorSigma[1, n], n]
dns = Select[DuffianQ][Range[1000000]];
Take[dns, UpTo[50]]
triplets = ToString[dns[[#]]] <> "\[LongDash]" <> ToString[dns[[# + 2]]] & /@ SequencePosition[Differences[dns], {1, 1}][[All, 1]]
Multicolumn[triplets, {Automatic, 5}, Appearance -> "Horizontal"]