A powerful number is a positive integer 
such that 
for every prime number divisor 
of 
. There are infinitely many powerful numbers, and the first
few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694).
Powerful numbers are always of the form 
for positive integers 
and 
.
The numbers of powerful numbers 
, 
, 
, ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044,
67231, 214122, 680330, 2158391, ... (OEIS A118896).
Golomb (1970) showed that the sum over the reciprocals of the powerful numbers 
is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
(OEIS A082695), where 
is the Riemann zeta
function.
Not every natural number is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently large natural number is the sum of at most three powerful numbers. There are infinitely many pairs of consecutive powerful numbers, the first few being (8, 9), (288, 289), (675, 676), (9800, 9801), ... (OEIS A060355 and A078326).
A cube-full number is a powerful number in which every prime number divisor occurs to exponent at least 3. Beyer de Ryke (2026) proved that the positive integers not representable as sums of at most three cube-full numbers have positive lower natural density.
Erdős (1976) conjectured that there do not exist three consecutive powerful numbers. Golomb (1970) also considered this question, as did Mollin and Walsh (1986). The conjecture remains open. It implies that there are infinitely many non-Wieferich primes (Granville 1986; Ribenboim 1989, p. 341; Vardi 1991).
Two restricted cases with a cubic number as the middle number have been proved. Chan (2025) showed that the system of equations
 |  |  |
(3)
|
 |  |  |
(4)
|
has no solution with prime numbers 
and 
and positive integers 
, 
, and 
. She (2025) showed that the system
of equations
 |  |  |
(5)
|
 |  |  |
(6)
|
has no solution with prime numbers 
and 
and positive integers 
, 
, and 
.
The term powerful number is also used for handsome numbers, which are sums of powers of their digits with positive integer exponents chosen independently for each digit.
