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:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $\map p n$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $25$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $168$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $1229$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $9592$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $78 \, 498$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $664 \, 579$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $5 \, 761 \, 455$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $50 \, 847 \, 534$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $455 \, 052 \, 511$
 * }
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $5 \, 761 \, 455$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $50 \, 847 \, 534$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $455 \, 052 \, 511$
 * }
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $455 \, 052 \, 511$
 * }