6 added 64 characters in body
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C#, 188 174 187 136 bytes

Golfed version thanks to TheLethalCoder for his great code golfing and tips (Try online!):

n=>{for(long a,b,c=3;n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b‌​*b*b==c*c*c+1)System‌​.Console.WriteLin‌e(‌​c*c*(a=c)+n/n--);};

Execution time to find the first 10 numbers: 33,370842 seconds on my i7 laptop (original version below was 9,618127 seconds for the same task).

Ungolfed version:

using System;

public class Program
{
    public static void Main()
    {
        Action<int> action = n =>
        {
            for (long a, b, d, c = 3; n > 0; c++)
                for (a = 2; a < c; a++)
                    for (b = a; b < c; b++)
                        if (a * a * a + b‌ * b * b == c * c * c + 1)
                            System‌.Console.WriteLin‌e( c * c * (a = c) + n / n--);
        };

        //Called like
        action(5);
    }
}

Previous golfed 187 bytes version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous (original) golfed 188 bytes version (Try online!):

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Execution time to find the first 10 numbers: 9,618127 seconds on my i7 laptop.

This is my first ever attempt in C# coding... A bit verbose compared to other languages...

C#, 188 174 187 136 bytes

Golfed version thanks to TheLethalCoder for his great code golfing and tips (Try online!):

n=>{for(long a,b,c=3;n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b‌​*b*b==c*c*c+1)System‌​.Console.WriteLin‌e(‌​c*c*(a=c)+n/n--);};

Execution time to find the first 10 numbers: 33,370842 seconds on my i7 laptop.

Ungolfed version:

using System;

public class Program
{
    public static void Main()
    {
        Action<int> action = n =>
        {
            for (long a, b, d, c = 3; n > 0; c++)
                for (a = 2; a < c; a++)
                    for (b = a; b < c; b++)
                        if (a * a * a + b‌ * b * b == c * c * c + 1)
                            System‌.Console.WriteLin‌e( c * c * (a = c) + n / n--);
        };

        //Called like
        action(5);
    }
}

Previous golfed 187 bytes version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous (original) golfed 188 bytes version (Try online!):

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Execution time to find the first 10 numbers: 9,618127 seconds on my i7 laptop.

This is my first ever attempt in C# coding... A bit verbose compared to other languages...

C#, 188 174 187 136 bytes

Golfed version thanks to TheLethalCoder for his great code golfing and tips (Try online!):

n=>{for(long a,b,c=3;n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b‌​*b*b==c*c*c+1)System‌​.Console.WriteLin‌e(‌​c*c*(a=c)+n/n--);};

Execution time to find the first 10 numbers: 33,370842 seconds on my i7 laptop (original version below was 9,618127 seconds for the same task).

Ungolfed version:

using System;

public class Program
{
    public static void Main()
    {
        Action<int> action = n =>
        {
            for (long a, b, d, c = 3; n > 0; c++)
                for (a = 2; a < c; a++)
                    for (b = a; b < c; b++)
                        if (a * a * a + b‌ * b * b == c * c * c + 1)
                            System‌.Console.WriteLin‌e( c * c * (a = c) + n / n--);
        };

        //Called like
        action(5);
    }
}

Previous golfed 187 bytes version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous (original) golfed 188 bytes version (Try online!):

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Execution time to find the first 10 numbers: 9,618127 seconds on my i7 laptop.

This is my first ever attempt in C# coding... A bit verbose compared to other languages...

5 added 442 characters in body
source | link

C#, 188 174 187187 136 bytes

Golfed version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes versionthanks to TheLethalCoder for his great code golfing and tips (thanks to Peter TaylorTry online!):

static void Main()n=>{for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++c=3;n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1a*a*a+b‌​*b*b==c*c*c+1)System‌​.Console.WriteLin‌​eWriteLin‌e(c*c*‌​c*c*(a=c)+n/n--);}

Previous golfed 188 bytes version:

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}};

Try online!

This is my first ever attempt in C# coding... A bit verbose compared to other languages, but takes around 10 secondsExecution time to elaboratefind the first 10 numbers: 33,370842 seconds on my i7 laptop.

using System;

public class Program
{
    public static void Main()
    {
        Action<int> action = n =>
        {
            for (long a, b, c=3d, n=int.Parse(Console.ReadLine());c n>0;= 3; n > 0; c++)
    {
            for (a = 2; a < c; a++)
        {
            for (b = a; b < c; b++)
            {
                if (a * a * a + bb‌ * b * b == c * c * c + 1)
                {
                    System‌.Console.WriteLin‌​eWriteLin‌e( c * c * (a = c) + n / n--);
        };

        //Called }like
        action(5);
    }
        }

Previous golfed 187 bytes version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}
 

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.WriteLineReadLine("End!"));n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous (original) golfed 188 bytes version (Try online!):

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadKeyReadLine();
);for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Execution time to find the first 10 numbers: 9,618127 seconds on my i7 laptop.

This is my first ever attempt in C# coding... A bit verbose compared to other languages...

C#, 188 174 187 bytes

Golfed version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 188 bytes version:

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Try online!

This is my first ever attempt in C# coding... A bit verbose compared to other languages, but takes around 10 seconds to elaborate the first 10 numbers on my i7 laptop.

static void Main()
{
    for (long a, b, c=3, n=int.Parse(Console.ReadLine()); n>0; c++)
    {
        for (a = 2; a < c; a++)
        {
            for (b = a; b < c; b++)
            {
                if (a * a * a + b * b * b == c * c * c + 1)
                {
                    Console.WriteLin‌​e(c * c * (a = c) + n / n--);
                }
            }
        }
    }
    Console.WriteLine("End!");
    Console.ReadKey();
}

C#, 188 174 187 136 bytes

Golfed version thanks to TheLethalCoder for his great code golfing and tips (Try online!):

n=>{for(long a,b,c=3;n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b‌​*b*b==c*c*c+1)System‌​.Console.WriteLin‌e(‌​c*c*(a=c)+n/n--);};

Execution time to find the first 10 numbers: 33,370842 seconds on my i7 laptop.

using System;

public class Program
{
    public static void Main()
    {
        Action<int> action = n =>
        {
            for (long a, b, d, c = 3; n > 0; c++)
                for (a = 2; a < c; a++)
                    for (b = a; b < c; b++)
                        if (a * a * a + b‌ * b * b == c * c * c + 1)
                            System‌.Console.WriteLin‌e( c * c * (a = c) + n / n--);
        };

        //Called like
        action(5);
    }
}

Previous golfed 187 bytes version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous (original) golfed 188 bytes version (Try online!):

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Execution time to find the first 10 numbers: 9,618127 seconds on my i7 laptop.

This is my first ever attempt in C# coding... A bit verbose compared to other languages...

4 deleted 5 characters in body
source | link

C#, 188 176 189174 187 bytes

Golfed version including using System;

using System;static void Main(){for(long a,b,d,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 176174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,d,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 188 bytes version:

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Try online!

This is my first ever attempt in C# coding... A bit verbose compared to other languages, but takes around 10 seconds to elaborate the first 10 numbers on my i7 laptop.

Ungolfed version:

static void Main()
{
    for (long a, b, d, c=3, n=int.Parse(Console.ReadLine()); n>0; c++)
    {
        for (a = 2; a < c; a++)
        {
            for (b = a; b < c; b++)
            {
                if (a * a * a + b * b * b == c * c * c + 1)
                {
                    Console.WriteLin‌​e(c * c * (a = c) + n / n--);
                }
            }
        }
    }
    Console.WriteLine("End!");
    Console.ReadKey();
}

C#, 188 176 189 bytes

Golfed version including using System;

using System;static void Main(){for(long a,b,d,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 176 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,d,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 188 bytes version:

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Try online!

This is my first ever attempt in C# coding... A bit verbose compared to other languages, but takes around 10 seconds to elaborate the first 10 numbers on my i7 laptop.

Ungolfed version:

static void Main()
{
    for (long a, b, d, c=3, n=int.Parse(Console.ReadLine()); n>0; c++)
    {
        for (a = 2; a < c; a++)
        {
            for (b = a; b < c; b++)
            {
                if (a * a * a + b * b * b == c * c * c + 1)
                {
                    Console.WriteLin‌​e(c * c * (a = c) + n / n--);
                }
            }
        }
    }
    Console.WriteLine("End!");
    Console.ReadKey();
}

C#, 188 174 187 bytes

Golfed version including using System;

using System;static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 174 bytes version (thanks to Peter Taylor):

static void Main(){for(long a,b,c=3,n=int.Parse(Console.ReadLine());n>0;c++)for(a=2;a<c;a++)for(b=a;b<c;b++)if(a*a*a+b*b*b==c*c*c+1)Console.WriteLin‌​e(c*c*(a=c)+n/n--);}

Previous golfed 188 bytes version:

static void Main(){double a,b,c,d;int t=0,n=Convert.ToInt32(Console.ReadLine());for(c=3;t<n;c++)for(a=2;a<c;a++)for(b=a;b<c;b++){d=(c*c*c)+1;if(a*a*a+b*b*b==d){Console.WriteLine(d);t++;}}}

Try online!

This is my first ever attempt in C# coding... A bit verbose compared to other languages, but takes around 10 seconds to elaborate the first 10 numbers on my i7 laptop.

Ungolfed version:

static void Main()
{
    for (long a, b, c=3, n=int.Parse(Console.ReadLine()); n>0; c++)
    {
        for (a = 2; a < c; a++)
        {
            for (b = a; b < c; b++)
            {
                if (a * a * a + b * b * b == c * c * c + 1)
                {
                    Console.WriteLin‌​e(c * c * (a = c) + n / n--);
                }
            }
        }
    }
    Console.WriteLine("End!");
    Console.ReadKey();
}
3 added 488 characters in body
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2 added 1 character in body
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1
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