An integer 
such that if 
, then 
, is called a powerful number. There are an infinite number
of 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 
.
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 A118893).
Erdős (1975/1965) conjectured that there do not exist three consecutive powerful numbers. Golomb (1970) also considered this question, as did Mollin and Walsh (1986). The conjecture that there are no powerful number triples implies that there are infinitely many non-Wieferich primes (Granville 1986; Ribenboim 1989, p. 341; Vardi 1991).
A separate usage of the term powerful number is for numbers which are the sums of any positive powers of their digits (not necessarily the same for each digit). The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, ... (OEIS A007532). These are also called handsome numbers by Rivera, and are a special case of the narcissistic numbers. Powerful numbers representable in two distinct ways (not counting different powers of duplicated digits as distinct) are 264, 373, 375, 2132, 2223, 2241, 2243, 2245, 2263, (OEIS A050240). Powerful numbers representable in two distinct ways (counting different powers of duplicated digits as distinct) are 224, 226, 264, 332, 334, 375, 377, 445, (OEIS A050241).
