[THOUGHT] I'll derive the integral formula symbolically, then use Python to search for the natural number n such that the integral from a_n to a_{n+1} equals the given value. [/THOUGHT] [PYTHON] from fractions import Fraction

target = Fraction(1407374833221632, 3)

def I_n(n): # I_n = (5/3)2^{2n-2} - 2^{n-1} from math import pow return Fraction(5,3) * (2*(2*n-2)) - 2**(n-1)

solutions = [] for n in range(1, 200): if I_n(n) == target: solutions.append(n)

print("Solutions n:", solutions)

Also directly solve 2^{n-1}(5*2^{n-1}-3) = 1407374833221632

M = 1407374833221632 candidates = [] for n in range(1, 200): k = 2**(n-1) if k*(5*k - 3) == M: candidates.append(n) print("Candidates from direct equation:", candidates)

[THOUGHT] The Python computation found that the unique natural number satisfying the integral condition is n=25. [/THOUGHT] \boxed{25}