# 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 divisor sums which are strictly decreasing:
$\map {\sigma_1} {44} = 84$, $\map {\sigma_1} {45} = 78$, $\map {\sigma_1} {46} = 72$, $\map {\sigma_1} {47} = 48$

The $2$nd of the $1$st pair of consecutive integers which both have $6$ divisors:
$\map {\sigma_0} {44} = \map {\sigma_0} {45} = 6$

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 $1$, $(10)$:
$45 = \ds \sum_{k \mathop = 1}^{3^2} k = \dfrac {3^2 \paren {3^2 + 1} } 2$

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

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

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

The $8$th odd positive integer after $1$, $3$, $5$, $7$, $9$, $15$, $21$ such that all smaller odd integers greater than $1$ which are coprime to it are prime.

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

The $22$nd odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $\ldots$

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:
$45 \to 09 \to 81 \to 63 \to 27 \to 45$