# 44

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## Number

$44$ (forty-four) is:

$2^2 \times 11$

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

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

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 $20$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.

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:
$\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$
$\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$
$\set {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}$
$\set {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}$