Happy & Special Numbers
Check for happy numbers, perfect numbers, and amicable pairs. Find happy numbers in any range.
19 is a HAPPY number!
Sequence:
19 → 82 → 68 → 100 → 1
19 is not a perfect number
Perfect numbers equal the sum of their proper divisors (6, 28, 496...)
19 is not part of an amicable pair
Amicable pairs: each equals sum of other's divisors (220 ↔ 284)
What are Happy Numbers?
A happy number is found by replacing it with the sum of the squares of its digits, then repeating until you reach 1 (happy) or enter a cycle (sad).
Example: 19 is Happy
- 1² + 9² = 1 + 81 = 82
- 8² + 2² = 64 + 4 = 68
- 6² + 8² = 36 + 64 = 100
- 1² + 0² + 0² = 1 ✓
Perfect Numbers
A perfect number equals the sum of its proper divisors (excluding itself). The first few are 6, 28, 496, and 8128.
Amicable Numbers
Two numbers are amicable if each equals the sum of the other's proper divisors. The smallest pair is 220 and 284.
Special Numbers in Puzzles
These number properties appear in puzzles:
- Testing if coordinates are happy numbers
- Finding the nth happy number
- Perfect number themed puzzles
- Mathematical curiosity challenges