A275256 - OEIS

%I #30 Dec 25 2019 23:10:11

%S 1225,1540,2926,4005,5985,8856,9045,9801,11781,11935,12376,12496,

%T 12720,13041,14400,16401,17200,17226,17290,17865,18096,21528,21736,

%U 23001,23751,24220,24976,25425,26796,27000,27405,27951,29241,29316,29601,29646,30976,31465,31536

%N Numbers with a minimum of 6 polygonal roots, excluding itself.

%C The i-th k-gonal number is equal to ((k-2)*i^2-(k-4)*i)/2. Sequence lists numbers n which are k-gonal numbers with k < n in at least 6 ways. - _N. J. A. Sloane_, Jul 25 2016

%C All polygonal roots (R) can be calculated for each number by checking if any numbers less than N give an integer result from (((K - 2) * (N * N) - (K - 4) * N) / 2), where K is increased until the numbers returned are larger than our N.

%H Chai Wah Wu, <a href="/A275256/b275256.txt">Table of n, a(n) for n = 1..10000</a>

%e For n = 5 we have a(5) = 5985. 5985 has 6 polygonal roots, since 5985 is the 45th octagonal number, the 35th dodecagonal number, the 18th 41-gonal number, the 9th 168-gonal number, the fifth 600-gonal number, and the third 1996-gonal number.

%o (C#)

%o List<BigInteger> CurrentBases = new List<BigInteger>();

%o List<BigInteger> CurrentNexts = new List<BigInteger>();

%o private int interesting2NumberPolygons;

%o public int Interesting2NumberPolygons

%o {

%o     get

%o     {

%o         return interesting2NumberPolygons;

%o     }

%o     set

%o     {

%o         interesting2NumberPolygons = value;

%o         OnPropertyChanged("Interesting2NumberPolygons");

%o     }

%o }

%o private BigInteger interesting2Number;

%o public BigInteger Interesting2Number

%o {

%o     get

%o     {

%o         return interesting2Number;

%o     }

%o     set

%o     {

%o         interesting2Number = value;

%o         OnPropertyChanged("Interesting2Number");

%o     }

%o }

%o private string fileLocation = "C:/NumberGen/";

%o public string FileLocation

%o {

%o     get

%o     {

%o         return fileLocation;

%o     }

%o     set

%o     {

%o         fileLocation = value;

%o         Properties.Settings.Default.LastLocation = value;

%o         Properties.Settings.Default.Save();

%o         OnPropertyChanged("FileLocation");

%o     }

%o }

%o private void FindAllIntegers()

%o     {

%o         Interesting2Number = 0;

%o         Interesting2NumberPolygons = 0;

%o         CurrentBases = new List<BigInteger>();

%o         CurrentNexts = new List<BigInteger>();

%o         BigInteger i = 0;

%o         int j = 0;

%o         while(true)

%o         {

%o             bool Finished = false;

%o             int k = 3;

%o             while (!Finished)

%o             {

%o                 if (k >= CurrentBases.Count)

%o                 {

%o                     CurrentBases.Add(1);

%o                     CurrentNexts.Add(1);

%o                 }

%o                 else

%o                 {

%o                     if(CurrentNexts[k] < i)

%o                     {

%o                         CurrentBases[k]++;

%o                         CurrentNexts[k] = PolygonalNumber(CurrentBases[k], k);

%o                     }

%o                     if(CurrentBases[k] <= 3 && CurrentNexts[k] >= i)

%o                     {

%o                         Finished = true;

%o                     }

%o                     k++;

%o                 }

%o             }

%o             if(CurrentNexts.FindAll(Nexts => Nexts == i).Count >= 6)

%o             {

%o                 List<int> Results = Enumerable.Range(0, CurrentNexts.Count)

%o                                     .Where(ind => CurrentNexts[ind] == i)

%o                                     .ToList();

%o                 string Row = "";

%o                 Row += i + "," + Results.Count;

%o                 foreach(int Result in Results)

%o                 {

%o                     Row += "," + Result + "," + CurrentBases[Result];

%o                 }

%o                 using (StreamWriter ResultsWriter = File.AppendText(@FileLocation + "Interesting2Numbers.dat"))

%o                 {

%o                     ResultsWriter.WriteLine(Row);

%o                 }

%o                 if(Results.Count >= Interesting2NumberPolygons)

%o                 {

%o                     Interesting2NumberPolygons = Results.Count;

%o                     Interesting2Number = i;

%o                 }

%o             }

%o             if (i % 100 == 0)

%o             {

%o                 Worker.ReportProgress(0, i);

%o                 using (StreamWriter DropCatcher = File.CreateText(@FileLocation + "DropCatcher.dat"))

%o                 {

%o                     DropCatcher.WriteLine(i);

%o                 }

%o             }

%o             j++;

%o             i++;

%o         }

%o     }

%o     private BigInteger PolygonalNumber(BigInteger N, BigInteger Sides)

%o     {

%o         if (Sides < 3)

%o         {

%o             return BigInteger.Zero;

%o         }

%o         //TRI: (N^2+N)/2

%o         else if (Sides == 3)

%o         {

%o             return ((N * N + N) / 2);

%o         }

%o         //POLY: ((S-2)N^2-(S-4)N)/2

%o         else

%o         {

%o             return (((Sides - 2) * (N * N) - (Sides - 4) * N) / 2);

%o         }

%o     }

%o (Python)

%o A275256_list = []

%o for m in range(2,10**5):

%o     n, c = 3, 0

%o     while (n*(n+1)) <= 2*m:

%o         if not 2*(n*(n-2) + m) % (n*(n - 1)):

%o             c += 1

%o             if c >= 6:

%o                 break

%o         n += 1

%o     if c >= 6:

%o         A275256_list.append(m) # _Chai Wah Wu_, Jul 25 2016

%K nonn

%O 1,1

%A _Matthew Parker_, Jul 21 2016

%E a(22)-a(39) from _Chai Wah Wu_, Jul 24 2016