Powerful Number

An Integer $m$ such that if $p\vert m$, then $p^2\vert m$, is called a powerful number. The first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (Sloane's A001694). Powerful numbers are always of the form $a^2b^3$ for $a,b\geq 1$.

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, but Erdös has conjectured that there do not exist three consecutive powerful numbers. The Conjecture that there are no powerful number triples implies that there are infinitely many Wieferich primes (Granville 1986, Vardi 1991).

A separate usage of the term powerful number is for numbers which are the sums of the positive powers of their digits. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, ... (Sloane's A007532).

References

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.

Ribenboim, P. ``Catalan's Conjecture.'' Amer. Math. Monthly 103, 529-538, 1996.

Sloane, N. J. A. Sequences A001694/M3325 and A007532/M0487 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 59-62, 1991.