Palindrome and Prime

93639 (a palindromic number) plus the sum of its prime factors is equal to 93683, which is a prime number:

93683 = 93639 + 3 + 7 + 7 + 7 + 7 + 13

Interestingly, there is no prime number between 93639 and 93683.

And it Continues…

(1⁵ + 2⁵) + (1⁷ + 2⁷) = 2 (1 + 2)⁴

(1⁵ + 2⁵ + 3⁵) + (1⁷ + 2⁷ + 3⁷) = 2 (1 + 2 + 3)⁴

(1⁵ + 2⁵ + 3⁵ + 4⁵) + (1⁷ + 2⁷ + 3⁷ + 4⁷) = 2 (1 + 2 + 3 + 4)⁴

(1⁵ + 2⁵ + 3⁵ + 4⁵ + 5⁵) + (1⁷ + 2⁷ + 3⁷ + 4⁷ + 5⁷) = 2 (1 + 2 + 3 + 4 + 5)⁴

etc.

Power Relationships

6205 = 38² + 69² and 3869 = 62² + 05²

5965 = 77² + 06² and 7706 = 59² + 65²

The following is less obvious and it involves cubes instead of squares:

244 = 1³ + 3³ + 6³ and 136 = 2³ + 4³ + 4³