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A powerful number is any positive integer whose prime factorization is composed only of powers. To put it another way, a powerful number has no prime number appearing only once in its prime factorization. Obviously, this set excludes all primes and most composite numbers and has infinitely many elements. A few examples of powerful numbers:

    8 = 23
    36 = 22 ∗ 32
    2,700 = 22 ∗ 33 ∗ 52

There are infinitely many consecutive pairs but whether any consecutive trios exist is an open question in mathematics. All powerful numbers are either Achilles numbers or perfect powers. The On-Line Encyclopedia of Integer Sequences list for the powerful numbers can be found here.

IRON NODER XIV: THE RETURN OF THE IRON NODER