Numbers of Primes with at most n Digits

Theorem

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

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

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

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

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

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $455,052,511$
• 1989: Paulo Ribenboim: The Book of Prime Number Records (2nd ed.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $168$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $50,847,534$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $455,052,511$