To find the 74th number in the sequence based on the pattern involving the sum of exponents of powers of 2, we need to generalize the pattern we've identified.

Generalizing the Pattern

The identified pattern is that each term in the sequence can be represented as a sum of powers of 2, where the sum of the exponents follows a specific arithmetic progression.

Given our previous observations:

• $a_5 = 21 = 2^4 + 2^2 + 2^0$ (sum of exponents is 6)
• $a_6 = 49 = 2^5 + 2^4 + 2^0$ (sum of exponents is 9)
• $a_7 = 76 = 2^6 + 2^3 + 2^2$ (sum of exponents is 11)
• $a_8 = 224 = 2^7$ (sum of exponents is 7)
• $a_9 = 514 = 2^9 + 2^1$ (sum of exponents is 10)

We observe that the sum of exponents sequence isn't immediately clear in a simple arithmetic progression. However, for simplicity, we can assume an underlying pattern based on previous examples and focus on calculating the exponents sums based on sequence behavior.

Formula and Calculation

If we assume the sum of exponents increases in a specific manner:

1. First, calculate a pattern for the sum of exponents up to $a_{74}$.

Given the complexity, let's hypothesize a simpler relationship based on previous sums and calculate:

$S(n) = a + b(n-1)$

We can assume an arithmetic pattern for sum increases, e.g., for each increment by a specific amount (like 1, 2, 3...).

Calculating $a_{74}$:

For practical purposes, let's assume an approximate linear progression for the sum of exponents:

If we calculate linearly: $S(n) = a + b(n-1)$

where $a$ and $b$ might be derived from:

$a_5: 6, a_6: 9, a_7: 11$

Pattern Derivation

• Let's assume $S(1) = 0$ (Base pattern origin)
• Incrementally it follows:

General increment: $S(n) = 6 + (n-1) \times 3$

For 74th term, apply: $S(74) = 6 + (74-1) \times 3 = 6 + 219 = 225$

Conversion to Value:

$2^{S(74)}$

Hence: $a_{74} \approx 2^{225}$

Conclusion

By this derived pattern, the 74th term's exponent sum approximates very large values involving $2^{225}$.

This approximation shows it might diverge simpler derivation and would require further in-depth specific formula confirmation: If considered simpler fraction approach: Would derive high computation and not fully clear manually validating exact previous sums.

For accurate confirmation: Would require practical computational tool for large sum validations.

Would you like detailed mathematical proof or more terms analysis for improved accuracy?

Questions for further:

1. Would you like exact computational validation tool for 74th term?
2. Do you prefer detailed verification pattern confirming each term sum exponents?
3. Could we simplify progression terms computation manually further?
4. Shall we derive additional sequence terms verification manually?
5. Want explanation deriving general exponent sum more accurately simplified?

Tip:

For complex series, break down patterns practically and computational for clarity ensuring accurate derivation simplifying terms relations better understanding.