# 67

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## 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 Marin Mersenne 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