67

Number
$67$ (sixty-seven) is:


 * The $19$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$


 * The $4$th tri-automorphic number after $2$, $5$, $7$:
 * $67^2 \times 3 = 13 \, 4 \mathbf {67}$


 * The $6$th prime number after $2$, $3$, $5$, $7$, $23$ consisting of a string of consecutive ascending digits


 * The $10$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$ such that the Mersenne number $2^p - 1$ is composite


 * The index of the $10$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$ which asserted to be prime
 * (in this case he was not correct: he missed $61$, and $2^{67} - 1 = 193 \, 707 \, 721 \times 761 \, 838 \, 257 \, 287$)


 * The $13$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$ such that no factorial of an integer can end with $n$ zeroes


 * The $16$th lucky number:
 * $1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $\ldots$


 * The $24$th positive integer which is not the sum of $1$ or more distinct squares:
 * $2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $\ldots$


 * The $29$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $\ldots$


 * The $31$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
 * $2^{67} = 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 928$


 * The $39$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$, $66$ which cannot be expressed as the sum of distinct pentagonal numbers


 * The $41$st (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $55$, $60$, $61$, $65$, $66$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

Also see

 * Factors of Mersenne Number M67