# 44

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## Number

$44$ (forty-four) is:

$2^2 \times 11$

The $5$th subfactorial after $0$, $1$, $2$, $9$:
$44 = 5! \left({1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} - \dfrac 1 {5!} }\right)$

The $10$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$:
$44 \to 4^2 + 4^2 = 16 + 16 = 32 \to 3^2 + 2^2 = 9 + 4 = 13 \to 1^2 + 3^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$

The length of the shortest edge of the smallest cuboid whose edges and the diagonals of whose faces are all integers:
The lengths of the edges are $44, 117, 240$
The lengths of the diagonals of the faces are $125, 244, 267$.

The largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset:
$\left\{ {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}\right\}$
$\left\{ {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}\right\}$
$\left\{ {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}\right\}$
$\left\{ {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}\right\}$

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

The $20$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $\ldots$

The $27$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$, $37$, $38$, $42$, $43$ which cannot be expressed as the sum of distinct pentagonal numbers.