Powerful Number
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 ![<span style=]()
{P_k}" src="https://mathworld.wolfram.com/images/equations/PowerfulNumber/Inline9.gif" style="height:15px; width:25px" /> is given by
(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).
REFERENCES:
Erdős, P. "Problems and Results on Consecutive Integers." Eureka 38, 3-8, 1975/1976.
Erdős, P. "Problems and Results on Consecutive Integers." Publ. Math. Debrecen 23, 271-282, 1976.
Golomb, S. W. "Powerful Numbers." Amer. Math. Monthly 77, 848-855, 1970.
Granville, A. "Powerful Numbers and Fermat's Last Theorem." C. R. Math. Rep. Acad. Sci. Canada 8, 215-218, 1986.
Guy, R. K. "Powerful Numbers." §B16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 67-73, 1994.
Heath-Brown, D. R. "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In Séminaire de Theorie des Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA: Birkhäuser, pp. 137-163, 1988.
Mollin, R. A. "The Power of Powerful Numbers." Int. J. Math. Math. Sci. 10, 125-130, 1986. https://www.math.ucalgary.ca/~ramollin/PPN.pdf.
Mollin, R. and Walsh, P. "On Powerful Numbers." Int. J. Math. Math. Sci. 9, 801-806, 1986.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1989.
Ribenboim, P. "Catalan's Conjecture." Amer. Math. Monthly 103, 529-538, 1996.
Rivera, C. "Problems & Puzzles: Puzzle 015-Narcissistic and Handsome Primes." https://www.primepuzzles.net/puzzles/puzz_015.htm.
Sloane, N. J. A. Sequences A001694/M3325, A007532/M0487, A050240, A050241, A060355, A082695, A118893, and A118896 in "The On-Line Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 59-62, 1991.
