This site is supported by donations to The OEIS Foundation.

# 24

Please do not rely on any information it contains.

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 ${\displaystyle 3x+1}$ sequence beginning at 24 24, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... ${\displaystyle 5x+1}$ 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 ${\displaystyle {\sqrt {24}}=2{\sqrt {6}}}$ and ${\displaystyle {\sqrt[{3}]{24}}=2{\sqrt[{3}]{3}}}$.

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

## Logarithms and 24th powers

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

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

 ${\displaystyle \log _{24}2}$ 0.218104 A152901 ${\displaystyle \log _{2}24}$ 4.58496 A155921 2 24 16777216 ${\displaystyle \log _{24}e}$ 0.314658 ${\displaystyle \log 24}$ 3.17805 A016647 ${\displaystyle e^{24}}$ ${\displaystyle \log _{24}3}$ 0.345687 A153100 ${\displaystyle \log _{3}24}$ 2.89279 A155922 3 24 282429536481 ${\displaystyle \log _{24}4}$ 0.436209 A153200 ${\displaystyle \log _{4}24}$ 2.29248 A155936 4 24 281474976710656 ${\displaystyle \log _{24}5}$ 0.506422 A153458 ${\displaystyle \log _{5}24}$ 1.97464 A155958 5 24 59604644775390625 ${\displaystyle \log _{24}6}$ 0.563791 A153614 ${\displaystyle \log _{6}24}$ 1.77371 A155959 6 24 4738381338321616896 ${\displaystyle \log _{24}7}$ 0.612296 A153736 ${\displaystyle \log _{7}24}$ 1.6332 A155964 7 24 191581231380566414401 ${\displaystyle \log _{24}8}$ 0.654313 A154007 ${\displaystyle \log _{8}24}$ 1.52832 A155975 8 24 4722366482869645213696 ${\displaystyle \log _{24}9}$ 0.691374 A154116 ${\displaystyle \log _{9}24}$ 1.44639 A155976 9 24 79766443076872509863361 ${\displaystyle \log _{24}10}$ 0.724527 A154174 ${\displaystyle \log _{10}24}$ 1.38021 A155979 10 24 1000000000000000000000000 ${\displaystyle \log _{24}11}$ 0.754517 A154195 ${\displaystyle \log _{11}24}$ 1.32535 A155981 11 24 9849732675807611094711841 ${\displaystyle \log _{24}12}$ 0.781896 A154216 ${\displaystyle \log _{12}24}$ 1.27894 A155982 12 24 79496847203390844133441536

See A010812 for the 24th powers of integers.

## Values for number theoretic functions with 24 as an argument

 ${\displaystyle \mu (24)}$ 0 ${\displaystyle M(24)}$ −4 ${\displaystyle \pi (24)}$ 9 ${\displaystyle \sigma _{1}(24)}$ 60 ${\displaystyle \sigma _{0}(24)}$ 8 ${\displaystyle \phi (24)}$ 8 ${\displaystyle \Omega (24)}$ 4 ${\displaystyle \omega (24)}$ 2 ${\displaystyle \lambda (24)}$ 2 This is the Carmichael lambda function. ${\displaystyle \lambda (22)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (24)}$ 1.000000059608189... 24! 620448401733239439360000 ${\displaystyle \Gamma (24)}$ 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 ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$, the factorization ${\displaystyle (2-2{\sqrt {-5}})(2+2{\sqrt {-5}})}$ is readily derived from the factorization ${\displaystyle (1-{\sqrt {-5}})(1+{\sqrt {-5}})}$.

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

## 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).

 ${\displaystyle -1}$ 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