Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value/Mistake

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$100$

Mistake

The largest number for which the sum of the primes less than the number of primes less than or equal to the number is the number itself. In this case, $\map \pi n = 25$, and the sum of the primes from $2$ to $23 = 100$. The other numbers with this property are $5$, $17$, $41$ and $77$.

Correction

The example given should have been presented as:

$\map \pi {100} = 25$

But note that for $n = 5$ and $n = 17$, the sum of the primes less than $\map \pi 5 = 3$ and $\map \pi {17} = 7$ falls short of $5$ and $17$.

$3$ and $7$ themselves need to be included in that sum.

Hence the statement should be amended to:

The largest number for which the sum of the primes less than or equal to the number of primes less than or equal to the number is the number itself.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $100$