45

Number
$45$ (forty-five) is:


 * $3^2 \times 5$


 * The $9$th triangular number after $1, 3, 6, 10, 15, 21, 28, 36$:
 * $45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = \dfrac {9 \times \left({9 + 1}\right)} 2$


 * The $5$th hexagonal number after $1, 6, 15, 28$:
 * $45 = 1 + 5 + 9 + 13 + 17 = 5 \left({2 \times 5 - 1}\right)$


 * The $2$nd hexamorphic number after $1$:
 * $45 = H_5$


 * The $3$rd Kaprekar number after $1, 9$:
 * $45^2 = 2025 \to 20 + 25 = 45$


 * The $3$rd Kaprekar triple after $1, 8$:
 * $45^3 = 91 \, 125 \to 9 + 11 + 25 = 45$


 * The $3$rd term of the $3$-Göbel sequence after $1, 2, 5$:
 * $45 = \left({1 + 1^3 + 2^3 + 5^3}\right) / 3$


 * The $2$nd of the $1$st ordered quadruple of consecutive integers that have sigma values which are strictly decreasing:
 * $\sigma \left({44}\right) = 84$, $\sigma \left({45}\right) = 78$, $\sigma \left({46}\right) = 72$, $\sigma \left({47}\right) = 48$


 * The $2$nd of the $1$st pair of consecutive integers which both have $6$ divisors:
 * $\tau \left({44}\right) = \tau \left({45}\right) = 6$


 * The $5$th positive integer $n$ after $4, 7, 15, 21$ such that $n - 2^k$ is prime for all $k$


 * The $28$th positive integer after $2, 3, 4, 7, 8, \ldots, 33, 37, 38, 42, 43, 44$ which cannot be expressed as the sum of distinct pentagonal numbers.

Also see