The next number in the sequence is 729.

Here's why:

The sequence represents the cubes of consecutive integers, but starting from 1 and skipping 4.

• $1^3 = 1$
• $2^3 = 8$
• $3^3 = 27$
• $5^3 = 125$
• $7^3 = 343$

The next integer to cube is 9:

• $9^3 = 729$

Thus, the next number is 729.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What are the cubes of all the numbers from 1 to 10?
2. Can you identify a pattern in the gaps between consecutive terms in the sequence?
3. How does this sequence differ from a standard consecutive cube sequence?
4. What would the sequence look like if it skipped even more numbers, say every other prime?
5. Can you describe a method for quickly recognizing a cube number?

Tip: Recognizing patterns in sequences often comes down to checking for basic mathematical operations like powers, multiplication, or primes!