# 45

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## Number

$45$ (forty-five) is:

$3^2 \times 5$

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 $2$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 total of all the entries in a magic square of order $3$, after $1$, $(10)$:
$45 = \displaystyle \sum_{k \mathop = 1}^{3^2} k = \dfrac {3^2 \paren {3^2 + 1} } 2$

The $4$th term of the $3$-Göbel sequence after $1$, $2$, $5$:
$45 = \left({1 + 1^3 + 2^3 + 5^3}\right) / 3$

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

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

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 $28$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.

One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:
$45 \to 09 \to 81 \to 63 \to 27 \to 45$