107

Number
$107$ (one hundred and seven) is:


 * The $28$th prime number


 * The $1$st of the $10$th pair of twin primes, with $109$


 * The $9$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$


 * The index of the $11$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$:
 * $M_{107} = 2^{107} - 1 = 162 \, 259 \, 276 \, 829 \, 213 \, 363 \, 391 \, 578 \, 010 \, 288 \, 127$


 * The $3$rd of the $1$st ordered triple of consecutive integers that have Euler $\phi$ values which are strictly increasing:
 * $\phi \left({105}\right) = 48$, $\phi \left({106}\right) = 52$, $\phi \left({107}\right) = 106$


 * The $4$th of the $2$nd ordered quadruple of consecutive integers that have sigma values which are strictly decreasing:
 * $\sigma \left({104}\right) = 210, \ \sigma \left({105}\right) = 192, \ \sigma \left({106}\right) = 162, \ \sigma \left({107}\right) = 108$


 * The $4$th prime number after $53, 71, 103$ which cannot be expressed as the difference between a power of $2$ and a power of $3$.


 * The $52$nd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$, $100$, $101$, $102$ which cannot be expressed as the sum of distinct pentagonal numbers.

Also see