A positive integer n is called powerful if for every prime p dividing n, p^2 also divides n; equivalently, in the prime factorization n = ∏ p_i^{α_i}, every exponent α_i ≥ 2.

Principle

Require that each prime factor appear with exponent at least two; powerful numbers are closed under multiplication by squares and are exactly the integers of the form a^2·b^3 where a and b are integers (not uniquely) in one canonical characterization.

Demonstration

72 = 2^3·3^2 is powerful because the primes dividing 72 are 2 and 3 and both squares 2^2 and 3^2 divide 72; exponents in factorization are 3 and 2, both ≥ 2.

Misapplication

Calling numbers with repeated prime factors 'powerful' when some prime appears only to the first power (e.g., 18 = 2·3^2 is not powerful because 2 appears to the first power), or conflating powerful numbers with perfect powers.

Consequence

Powerful numbers have simplified behavior under certain multiplicative arithmetic functions and appear in the study of integer solutions to equations where high prime multiplicity is required; they form a sparse but infinite subset of integers.

Reversal

The complementary set consists of integers that have at least one prime factor to the first power; reversing the condition permits primes and squarefree numbers rather than requiring squared primes.

Boundary

Definition applies only to positive integers and depends on prime factorization; numbers with a prime factor of exponent one are excluded. The representation as a^2·b^3 is not canonical and must be applied with care; powerfulness is stable under multiplication by any square.

Semantic Tension

Near the notion of 'squarefull' (which is a synonym in many contexts) and 'perfect power'; tension arises because 'squarefull' is sometimes used interchangeably while 'perfect power' imposes global exponent commonality rather than per-prime exponents ≥2.

Synthesis

A powerful number is a positive integer whose prime factorization has no exponent equal to one, i.e., every prime dividing the number divides it at least to the second power, yielding numbers rich in repeated prime factors.