Numbers of Primes with at most n Digits

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Theorem

Let $p: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:

$\forall n \in \Z_{>0}: p \left({n}\right) = $ the number of prime numbers with no more than $n$ digits

Then the value of $p$ for the first few numbers is given below:

$n$ $p \left({n}\right)$
$1$ $4$
$2$ $25$
$3$ $168$
$4$ $1229$
$5$ $9592$
$6$ $78 \, 498$
$7$ $664 \, 579$
$8$ $5 \, 761 \, 455$
$9$ $50 \, 847 \, 534$
$10$ $455 \, 052 \, 511$

This sequence is A006880 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Historical Note

The number of primes less than $10^{10}$ was calculated by Derrick Norman Lehmer.


Sources