Number of primes less than or equal to a number up to 3,000, approximation formula

[greg1313]

I discovered a relatively good approximation formula for $\displaystyle \pi(x), \$ which gives the exact number of prime numbers
less than or equal to the value of x. For example, $\displaystyle \ \pi(10) = 4, \$ because there are four prime numbers less than
or equal to 10. My formula works best in the interval of about $\displaystyle \ 10 \le x \le 3,000.$

Approximation formula: $\displaystyle \ \ Ceiling \ function \ of \$$\displaystyle \ \bigg[ \dfrac{x}{\log_{10}{(x^2)}} \bigg]$

I cherry-picked multiples of 25 for certain x-values, where each approximation formula's value equals the corresponding $\displaystyle \pi(x)$ value.

x $\displaystyle \ \ \ \ \ \ \$| Approx. formula value = $\displaystyle \ \pi(x)$
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$\displaystyle \ \ \$ 25 $\displaystyle \ \ \ \ \ \ \ \ \$ 9
$\displaystyle \ \ \$ 50 $\displaystyle \ \ \ \ \ \ \$ 15
$\displaystyle \$ 100 $\displaystyle \ \ \ \ \ \ \$ 25
$\displaystyle \$ 125 $\displaystyle \ \ \ \ \ \ \$ 30
$\displaystyle \$ 150 $\displaystyle \ \ \ \ \ \ \$ 35
$\displaystyle \$ 175 $\displaystyle \ \ \ \ \ \ \$ 40
$\displaystyle \$ 225 $\displaystyle \ \ \ \ \ \ \$ 48
$\displaystyle \$ 250 $\displaystyle \ \ \ \ \ \ \$ 53
$\displaystyle \$ 550 $\displaystyle \ \ \ \ \$ 101
$\displaystyle \$ 575 $\displaystyle \ \ \ \ \$ 105
$\displaystyle \$ 850 $\displaystyle \ \ \ \ \$ 146
$\displaystyle \$ 975 $\displaystyle \ \ \ \ \$ 164
1375 $\displaystyle \ \ \ \ \$ 220
1475 $\displaystyle \ \ \ \ \$ 233
1775 $\displaystyle \ \ \ \ \$ 274
1875 $\displaystyle \ \ \ \ \$ 287
1900 $\displaystyle \ \ \ \ \$ 290
2000 $\displaystyle \ \ \ \ \$ 303
2100 $\displaystyle \ \ \ \ \$ 317
2175 $\displaystyle \ \ \ \ \$ 326
2725 $\displaystyle \ \ \ \ \$ 397
2775 $\displaystyle \ \ \ \ \$ 403
2800 $\displaystyle \ \ \ \ \$ 407
2825 $\displaystyle \ \ \ \ \$ 410
2850 $\displaystyle \ \ \ \ \$ 413
2900 $\displaystyle \ \ \ \ \$ 419
2925 $\displaystyle \ \ \ \ \$ 422
2975 $\displaystyle \ \ \ \ \$ 429

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You may wish to experiment with this approximation formula on your own.