Power-full number

2020 Mathematics Subject Classification: Primary: 11A05 [MSN][ZBL]

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

[BG]	Paul T. Bateman, Emil Grosswald, "On a theorem of Erdős and Szekeres" Ill. J. Math. 2 (1958) 88-98 Zbl 0079.07104
[Fi]	Steven R. Finch, "Mathematical Constants", Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
[Gu]	Richard K. Guy, Unsolved Problems in Number Theory 3rd ed. Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001