Input a number to check if it's a happy number, or provide a range to generate all happy numbers within it.
A happy number is a positive integer that eventually reaches 1 after repeatedly replacing the number with the sum of the squares of its digits. If this process results in a cycle without reaching 1, the number is not a happy number.
Steps to Check:
Key Consideration: Keep track of sums that have already appeared to detect cycles. If a sum repeats, the number is not a happy number.
Solution:
Calculation Steps:
\( 4^2 + 7^2 + 8^2 = 16 + 49 + 64 = 129 \)
\( 1^2 + 2^2 + 9^2 = 1 + 4 + 81 = 86 \)
\( 8^2 + 6^2 = 64 + 36 = 100 \)
\( 1^2 + 0^2 + 0^2 = 1 + 0 + 0 = 1 \)
Result: Since the result is 1, 478 is a happy number.
Solution:
Calculation Steps:
\( 2^2 + 0^2 + 2^2 + 3^2 = 4 + 0 + 4 + 9 = 17 \)
\( 1^2 + 7^2 = 1 + 49 = 50 \)
\( 5^2 + 0^2 = 25 + 0 = 25 \)
\( 2^2 + 5^2 = 4 + 25 = 29 \)
\( 2^2 + 9^2 = 4 + 81 = 85 \)
\( 8^2 + 5^2 = 64 + 25 = 89 \)
\( 8^2 + 9^2 = 64 + 81 = 145 \)
\( 1^2 + 4^2 + 5^2 = 1 + 16 + 25 = 42 \)
\( 4^2 + 2^2 = 16 + 4 = 20 \)
\( 2^2 + 0^2 = 4 + 0 = 4 \)
\( 4^2 = 16 \)
\( 1^2 + 6^2 = 1 + 36 = 37 \)
\( 3^2 + 7^2 = 9 + 49 = 58 \)
\( 5^2 + 8^2 = 25 + 64 = 89 \)
The sum \(89\) appears again, indicating a cycle. Therefore, 2023 is not a happy number.