A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful. Powerful numbers are also known as squarefull, square-full, or 2-full. (Not to be confused with the term squareful, which refers to numbers that are not square-free.) The following is a list of all powerful numbers between 1 and 1000:
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... (sequence A001694 in the OEIS).
Equivalence of the two definitions If m = a2b3, then every prime in the prime factorization of a appears in the prime factorization of m with an exponent of at least two, and every prime in the prime factorization of b appears in the prime factorization of m with an exponent of at least three; therefore, m is powerful. In the other direction, suppose that m is powerful, with prime factorization
m = ∏ p i α i , {\displaystyle m=\prod p_{i}^{\alpha _{i}},}
where each αi ≥ 2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi − γi. Then, all values βi are nonnegative even integers, and all values γi are either zero or three, so
m = ( ∏ p i β i ) ( ∏ p i γ i ) = ( ∏ p i β i / 2 ) 2 ( ∏ p i γ i / 3 ) 3 {\displaystyle m=\left(\prod p_{i}^{\beta _{i}}\right)\left(\prod p_{i}^{\gamma _{i}}\right)=\left(\prod p_{i}^{\beta _{i}/2}\right)^{2}\left(\prod p_{i}^{\gamma _{i}/3}\right)^{3}}
supplies the desired representation of m as a product of a square and a cube. Informally, given the prime factorization of m, take b to be the product of the prime factors of m that have an odd exponent (if there are none, then take b to be 1). Because m is powerful, each prime factor with an odd exponent has an exponent that is at least 3, so m/b3 is an integer. In addition, each prime factor of m/b3 has an even exponent, so m/b3 is a perfect square, so call this a2; then m = a2b3. For example:
m = 21600 = 2 5 × 3 3 × 5 2 , {\displaystyle m=21600=2^{5}\times 3^{3}\times 5^{2}\,,}
b = 2 × 3 = 6 , {\displaystyle b=2\times 3=6\,,}
a = m b 3 = 2 2 × 5 2 = 10 , {\displaystyle a={\sqrt {\frac {m}{b^{3}}}}={\sqrt {2^{2}\times 5^{2}}}=10\,,}
m = a 2 b 3 = 10 2 × 6 3 . {\displaystyle m=a^{2}b^{3}=10^{2}\times 6^{3}\,.}
The representation m = a2b3 calculated in this way has the property that b is squarefree, and is uniquely defined by this property.
Mathematical properties The sum of the reciprocals of the powerful numbers converges. The value of this sum may be written in several other ways, including as the infinite product
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