24

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24 is an integer, the largest divisible by each integer less than its square root.

Membership in core sequences

Even numbers	..., 18, 20, 22, 24, 26, 28, 30, ...	A005843(12)
Composite numbers	..., 20, 21, 22, 24, 25, 26, 27, ...	A002808
Factorial numbers	1, 2, 6, 24, 120, 720, 5040, ...	A000142
Abundant numbers	12, 18, 20, 24, 30, 36, 40, 42, ...	A005101
Planar partition numbers	1, 1, 3, 6, 13, 24, 48, 86, 160, ...	A000219

Sequences pertaining to 24

Multiples of 24	0, 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, ...	A008606
Divisors of 24	1, 2, 3, 4, 6, 8, 12, 24	A018253
Squares modulo 24	0, 1, 4, 9, 12, 16	A010386
24-gonal numbers	0, 1, 24, 69, 136, 225, 336, 469, 624, 801, 1000, ...	A051876
Centered 24-gonal numbers	1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, ...	A069173
Concentric 24-gonal numbers	1, 24, 49, 96, 145, 216, 289, 384, 481, 600, 721, ...	A195158
$3x+1$ sequence beginning at 24	24, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
$5x+1$ sequence beginning at 24	24, 12, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, ...

Partitions of 24

There are 1575 partitions of 24.

The Goldbach representations of 24 are: 5 + 19 = 7 + 17 = 11 + 13.

Roots and powers of 24

In the table below, irrational numbers are given truncated to eight decimal places.

Do note that ${\sqrt {24}}=2{\sqrt {6}}$ and ${\sqrt[{3}]{24}}=2{\sqrt[{3}]{3}}$.

${\sqrt {24}}$	4.89897948	A010480	24 2	576
${\sqrt[{3}]{24}}$	2.88449914	A010596	24 3	13824
${\sqrt[{4}]{24}}$	2.21336383	A011020	24 4	331776
${\sqrt[{5}]{24}}$	1.88817502	A011109	24 5	7962624
${\sqrt[{6}]{24}}$	1.69838132		24 6	191102976
${\sqrt[{7}]{24}}$	1.57461010		24 7	4586471424
${\sqrt[{8}]{24}}$	1.48773782		24 8	110075314176
${\sqrt[{9}]{24}}$	1.42349781		24 9	2641807540224
${\sqrt[{10}]{24}}$	1.37410881		24 10	63403380965376
${\sqrt[{11}]{24}}$	1.33497689		24 11	1521681143169024
${\sqrt[{12}]{24}}$	1.30321960		24 12	36520347436056576
				A009968

Logarithms and 24th powers

In the OEIS specifically and mathematics in general, $\log x$ refers to the natural logarithm of $x$, whereas all other bases are specified with a subscript.

As above, irrational numbers in the following table are truncated to eight decimal places.

$\log _{24}2$	0.21810429	A152901	$\log _{2}24$	4.58496250	A155921	2 24	16777216
$\log _{24}e$	0.31465798		$\log 24$	3.17805383	A016647	$e^{24}$
$\log _{24}3$	0.34568712	A153100	$\log _{3}24$	2.89278926	A155922	3 24	282429536481
$\log _{24}4$	0.43620858	A153200	$\log _{4}24$	2.29248125	A155936	4 24	281474976710656
$\log _{24}5$	0.50642248	A153458	$\log _{5}24$	1.97463586	A155958	5 24	59604644775390625
$\log _{24}6$	0.56379141	A153614	$\log _{6}24$	1.77370561	A155959	6 24	4738381338321616896
$\log _{24}7$	0.61229615	A153736	$\log _{7}24$	1.63319659	A155964	7 24	191581231380566414401
$\log _{24}8$	0.65431287	A154007	$\log _{8}24$	1.52832083	A155975	8 24	4722366482869645213696
$\log _{24}9$	0.69137424	A154116	$\log _{9}24$	1.44639463	A155976	9 24	79766443076872509863361
$\log _{24}10$	0.72452677	A154174	$\log _{10}24$	1.38021124	A155979	10 24	1000000000000000000000000
$\log _{24}11$	0.75451688	A154195	$\log _{11}24$	1.32535138	A155981	11 24	9849732675807611094711841
$\log _{24}12$	0.78189570	A154216	$\log _{12}24$	1.27894294	A155982	12 24	79496847203390844133441536

See A010812 for the 24th powers of integers.

Values for number theoretic functions with 24 as an argument

$\mu (24)$	0
$M(24)$	−4
$\pi (24)$	9
$\sigma _{1}(24)$	60
$\sigma _{0}(24)$	8
$\phi (24)$	8
$\Omega (24)$	4
$\omega (24)$	2
$\lambda (24)$	2	This is the Carmichael lambda function.
$\lambda (22)$	1	This is the Liouville lambda function.
$\zeta (24)$	1.000000059608189...
24!	620448401733239439360000
$\Gamma (24)$	25852016738884976640000

Factorization of 24 in some quadratic integer rings

As was mentioned above, 24 is the product of 2 3 and 3. But it has different factorizations in some quadratic integer rings. In rings where it has more than one distinct factorization, the extra factorizations generally derive from the multiple factorizations of 6 in that ring. For example, in $\mathbb {Z} [{\sqrt {-5}}]$, the factorization $(2-2{\sqrt {-5}})(2+2{\sqrt {-5}})$ is readily derived from the factorization $(1-{\sqrt {-5}})(1+{\sqrt {-5}})$.

$\mathbb {Z} [i]$	$(1\pm i)^{3}3$
$\mathbb {Z} [{\sqrt {-2}}]$	$(-1)({\sqrt {-2}})^{6}(1\pm {\sqrt {-2}})$	$\mathbb {Z} [{\sqrt {2}}]$	$({\sqrt {2}})^{6}3$
$\mathbb {Z} [\omega ]$	$(-1)2^{3}(1+2\omega )^{2}$	$\mathbb {Z} [{\sqrt {3}}]$	$(1\pm {\sqrt {3}})^{3}({\sqrt {3}})^{2}$
$\mathbb {Z} [{\sqrt {-5}}]$	2 3 × 3	$\mathbb {Z} [\phi ]$	2 3 × 3
$\mathbb {Z} [{\sqrt {-6}}]$		$\mathbb {Z} [{\sqrt {6}}]$	$(-1)(2\pm {\sqrt {6}})^{3}(3\pm {\sqrt {6}})$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$	$\left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)^{3}3$	$\mathbb {Z} [{\sqrt {7}}]$	$(-1)(3\pm {\sqrt {7}})^{3}(2\pm {\sqrt {7}})$
$\mathbb {Z} [{\sqrt {-10}}]$	2 3 × 3	$\mathbb {Z} [{\sqrt {10}}]$	2 3 × 3
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$	$2^{3}\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)$	$\mathbb {Z} [{\sqrt {11}}]$	$(3\pm {\sqrt {11}})^{3}3$
$\mathbb {Z} [{\sqrt {-13}}]$	2 3 × 3	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$	$(-1)2^{3}\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)$
$\mathbb {Z} [{\sqrt {-14}}]$		$\mathbb {Z} [{\sqrt {14}}]$	$(4\pm {\sqrt {14}})^{3}3$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$		$\mathbb {Z} [{\sqrt {15}}]$	2 3 × 3
$\mathbb {Z} [{\sqrt {-17}}]$		${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$	$\left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)^{3}3$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$		$\mathbb {Z} [{\sqrt {19}}]$	$(13\pm 3{\sqrt {19}})^{3}(4\pm {\sqrt {19}})$

Representation of 24 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	11000	220	120	44	40	33	30	26	24	22	20	1B	1A	19	18	17	16	15	14

Note that 24 is a Harshad number in every base from binary to base 13. It is also a Harshad number in bases 17, 19, 21, 22, 23, and trivially so in bases 24 and higher, and in factorial base (see A118363).

See also

Some integers

							$-1$
0	1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
					1729