167

Number

$167$ (one hundred and sixty-seven) is:

The $3$rd of the $2$nd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ that have Euler $\phi$ values which are strictly increasing:

$\map \phi {165} = 80$, $\map \phi {166} = 82$, $\map \phi {167} = 166$

The $9$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$:

$167 = 2 \times 83 + 1$

The $10$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$, $157$, $163$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$

The $13$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$

The $14$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$

The $18$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$

The $27$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$, $109$, $129$, $130$, $133$, $139$:

$167 \to 1^2 + 6^2 + 7^2 = 1 + 36 + 49 = 86 \to 8^2 + 6^2 = 64 + 36 = 100 \to 1^2 + 0^2 + 0^2 = 1$

The $33$rd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.