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Divisibility-based sets of integers

Form of factorization:

Prime number

Composite number

Powerful number

Square-free number

Achilles number

Constrained divisor sums:

Perfect number

Almost perfect number

Quasiperfect number

Multiply perfect number

Hyperperfect number

Unitary perfect number

Semiperfect number

Primitive semiperfect number

Practical number

Numbers with many divisors:

Abundant number

Highly abundant number

Superabundant number

Colossally abundant number

Highly composite number

Superior highly composite number

Other:

Deficient number

Weird number

Amicable number

Sociable number

Sublime number

Harmonic divisor number

See also:

Divisor function

Divisor

Prime factor

Factorization

A powerful number is a positive integer m 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. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

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 OEIS)

Contents 1 Equivalence of the two definitions 2 Mathematical properties 3 Sums and differences of powerful numbers 4 Generalization 5 See also 6 References 7 External links

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

where each α_i ≥ 2. Define γ_i to be three if α_i is odd, and zero otherwise, and define β_i = α_i - γ_i. Then, all values β_i are nonnegative even integers, and all values γ_i are either zero or three, so

supplies the desired representation of m as a product of a square and a cube.

The representation m = a²b³ calculated in this way has the property that b is squarefree, and is uniquely defined by this property.

The sum of reciprocals of powerful numbers converges to

where p runs over all primes, ζ(s) denotes Riemann's zeta function, and ζ(3) is Apéry's constant (Golomb, 1970).

Let k(x) denote the number of powerful numbers in the interval [1,x]. Then k(x) is proportional to the square root of x. More precisely,

(Golomb, 1970).

The sequence of pairs of consecutive powerful numbers is given by (sequence A060355 in OEIS). The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation x² − 8y² = 1 has infinitely many integral solutions, there are infinitely many pairs of consecutive powerful numbers (Golomb, 1970); more generally, one can find consecutive powerful numbers by solving a similar Pell equation x² − ny² = ±1 for any perfect cube n. However, one of the two powerful numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are infinitely many pairs of consecutive powerful numbers such as (23³, 2³3²13²) in which neither number in the pair is a square. It is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers; if the abc conjecture is true, it would follow that there are only finitely many triples of consecutive powerful numbers.

Any odd number is a difference of two consecutive squares: 2k + 1 = (k + 1)² - k². Similarly, any multiple of four is a difference of the squares of two numbers that differ by two. However, a singly even number, that is, a number divisible by two but not by four, cannot be expressed as a difference of squares. This motivates the question of determining which singly even numbers can be expressed as differences of powerful numbers. Golomb exhibited some representations of this type:

2 = 3³ − 5²

10 = 13³ − 3⁷

18 = 19² − 7³ = 3²(3³ − 5²).

It had been conjectured that 6 cannot be so represented, and Golomb conjectured that there are infinitely many integers which cannot be represented as a difference between two powerful numbers. However, Narkiewicz showed that 6 can be so represented in infinitely many ways such as

6 = 5⁴7³ − 463²,

and McDaniel showed that every integer has infinitely many such representations(McDaniel, 1982).

Erdős conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by Roger Heath-Brown (1987).

More generally, we can consider the integers all of whose prime factors have exponents at least k. Such an integer is called a k-powerful number, k-ful number, or k-full number.

(2^k+1 − 1)^k,  2^k(2^k+1 − 1)^k,   (2^k+1 − 1)^k+1

are k-powerful numbers in an arithmetic progression. Moreover, if a₁, a₂, ..., a_s are k-powerful in an arithmetic progression with common difference d, then

a₁(a_s + d)^k,  

a₂(a_s + d)^k, ..., a_s(a_s + d)^k, (a_s + d)^k+1

are s + 1 k-powerful numbers in an arithmetic progression.

We have an identity involving k-powerful numbers:

a^k(a^l + ... + 1)^k + a^{k + 1}(a^l + ... + 1)^k + ... + a^{k + l}(a^l + ... + 1)^k = a^k(a^l + ... +1)^k+1.

This gives infinitely many l+1-tuples of k-powerful numbers whose sum is also k-powerful. Nitaj shows there are infinitely many solutions of x+y=z in relatively prime 3-powerful numbers(Nitaj, 1995). Cohn constructs an infinite family of solutions of x+y=z in relatively prime non-cube 3-powerful numbers as follows: the triplet

X = 9712247684771506604963490444281, Y = 32295800804958334401937923416351, Z = 27474621855216870941749052236511

is a solution of the equation 32X³ + 49Y³ = 81Z³. We can construct another solution by setting X′ = X(49Y³ + 81Z³), Y′ = −Y(32X³ + 81Z³), Z′ = Z(32X³ − 49Y³) and omitting the common divisor.

• Achilles number

• Cohn, J. H. E. (1998). "A conjecture of Erdős on 3-powerful numbers". Math. Comp. 67: 439–440. 

• Erdős, Paul and Szekeres, George (1934). "Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem". Acta Litt. Sci. Szeged 7: 95–102. 

• Golomb, Solomon W. (1970). "Powerful numbers". American Mathematical Monthly 77: 848–852. 

• Guy, Richard K. (2004). Unsolved Problems in Number Theory, 3rd edition. Springer-Verlag, Section B16. ISBN 0-387-20860-7. 

• Heath-Brown, Roger (1988). "Ternary quadratic forms and sums of three square-full numbers". Séminaire de Théorie des Nombres, Paris, 1986-7: 137–163, Boston: Birkhäuser. 

• Heath-Brown, Roger (1990). "Sums of three square-full numbers". Number Theory, I (Budapest, 1987): 163–171, Colloq. Math. Soc. János Bolyai, no. 51. 

• McDaniel, Wayne L. (1982). "Representations of every integer as the difference of powerful numbers". Fibonacci Quarterly 20: 85–87. 

• Nitaj, Abderrahmane (1995). "On a conjecture of Erdős on 3-powerful numbers". Bull. London Math. Soc. 27: 317–318. 

• Powerful number
• The abc conjecture