73

Number
$73$ (seventy-three) is:


 * The $21$st prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$, $71$


 * The only Sheldon prime.


 * The $1$st of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
 * $73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$


 * The $2$nd of the $8$th pair of twin primes, with $71$


 * The $1$st of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly increasing:
 * $\map {\sigma_1} {73} = 74$, $\map {\sigma_1} {74} = 114$, $\map {\sigma_1} {75} = 124$, $\map {\sigma_1} {76} = 140$


 * The lower end of the $5$th record-breaking gap between twin primes:
 * $101 - 73 = 28$


 * The $6$th emirp after $13$, $17$, $31$, $37$, $71$


 * The $8$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$:
 * $73$, $7$, $3$ are prime


 * The $9$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
 * $0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$


 * The $11$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$


 * The $12$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$, $67$, $71$ such that the Mersenne number $2^p - 1$ is composite


 * The $14$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$ such that no factorial of an integer can end with $n$ zeroes


 * The $17$th lucky number:
 * $1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $\ldots$


 * The $31$st odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $\ldots$


 * Every positive integer can be expressed as the sum of at most $73$ $6$th powers


 * In the smallest equilateral triangle with sides of integer length ($112$) which contains a point which is an integer distance from each vertex, the distance from that point to its furthest vertex (the other two being $57$ and $65$)

The Sheldon Cooper Exposition
It may be worth adding the following observation about the Hilbert-Waring Theorem:
 * Every positive integer can be expressed as the sum of at most $37$ positive $5$th powers.

and:
 * Every positive integer can be expressed as the sum of at most $73$ (positive) $6$th powers.

also notes:
 * $73 \times 37 = 2701$ is not only the $73$rd triangular number, but also the $37$th hexagonal number.

Also see

 * Sheldon Conjecture
 * 2701


 * Hilbert-Waring Theorem for $6$th Powers
 * Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices