Remember: Include and exclude!

Of our 10^6 starting values... There are 1000 values that are perfect squares. (1^2 to 1000^2) There are 100 values that are perfect cubes, (1^3 to 100^3) however 10 of these [ a perfect square that is then cubed] can also be expressed as square numbers. For example 4^3 = (2^2)^3 = (2^3)^2 = 8^2 so need to be excluded from the count as it has already been included in the line above, leaving 90 perfect cubes that are not square. Though there are 31 perfect forth powers, (1^4 to 31^4), each of these can be expressed as a perfect square, previously accounted for; so need not be considered. eg 31^4 = (31^2)^2 < 961^2

From the 1,000,000 starting integers 1090 have been identified as powers of 2, 3, or 4 leaving 998910 integers that are therefore not powers of 2, 3 or 4.

First, count how many integer perfect squares, cubes and fourth powers are there between 1 and 1000000. Counting with Mathematica into three lists:

$\text{squares}\text{:=}\{\}$

$\text{cubes}\text{:=}\{\}$

$\text{quadroes}\text{:=}\{\}$

$\text{For}\left[n=1,n\leq 10^6,n\text{++},\right.$

$k=\sqrt{n};$

$l=\sqrt[3]{n};$

$m=\sqrt[4]{n};$

$\text{If}[\text{IntegerQ}[k]=\text{True},\text{AppendTo}[\text{squares},n]];$

$\text{If}[\text{IntegerQ}[l]=\text{True},\text{AppendTo}[\text{cubes},n]];$

$\text{If}[\text{IntegerQ}[m]=\text{True},\text{AppendTo}[\text{quadroes},n]]]$

Then, to count how many of the integers between 1 to 1000000 are neither perfect squares nor perfect cubes nor perfect fourth powers, substract the union of the lists "squares", "cubes" and "quadroes" from the total number of integers between 1 and 1000000:

$1000000-\text{Length}[\text{squares}\cup \text{cubes}\cup \text{quadroes}]$ = $\boxed{998910}$