View source for Power-full number

{{TEX|done}}{{MSC|11A05}}

''of type $k$''

A natural number $n$ with the property that if a prime $p$ divides $n$, then $p^k$ divides $n$.  A square-full number is a power-full number of type 2; a cube-full number is a power-full number of type 3.

If $N_k(x)$ counts the $k$-full numbers $\le x$, then 
$$
N_2(x) = \frac{\zeta\left({\frac{3}{2}}\right) }{\zeta(3) } x^{\frac{1}{2}} + \frac{\zeta\left({\frac{2}{3}}\right) }{\zeta(2) } x^{\frac{1}{3}} + o\left({x^{\frac{1}{6}}}\right)
$$
where $\zeta(s)$ is the [[Riemann zeta function‏]].   Similarly,
$$
N_3(x) = c_{03} x^{\frac{1}{3}} + c_{13} x^{\frac{1}{4}} + c_{23} x^{\frac{1}{5}} + o\left({ x^{\frac{1}{8}} }\right) 
$$
and generally
$$
N_k(x) = c_{0k} x^{1/k} + O(x^{1/(k+1)}) 
$$
where
$$
c_{0k} = \prod_p \left({ 1 + \sum_{m=k+1}^{2k-1} p^{-m/k} }\right) \ .
$$
The $c_{\circ k}$ are the '''Bateman–Grosswald constants'''.

==References==
{|
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|valign="top"|{{Ref|BG}}||valign="top"|  Paul T. Bateman, Emil Grosswald, "On a theorem of Erdős and Szekeres" Ill. J. Math. '''2'''  (1958) 88-98 {{ZBL|0079.07104}}
|-
|valign="top"|{{Ref|Fi}}||valign="top"|  Steven R. Finch, "Mathematical Constants", Cambridge University Press (2003) {{ISBN|0-521-81805-2}} {{ZBL|1054.00001}}
|-
|valign="top"|{{Ref|Gu}}||valign="top"|  Richard K. Guy, ''Unsolved Problems in Number Theory'' 3rd ed. Springer-Verlag (2004) {{ISBN|0-387-20860-7}} {{ZBL|1058.11001}}
|}