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# 32

Please do not rely on any information it contains.

32 is the fifth power of 2.

## Membership in core sequences

 Even numbers ..., 26, 28, 30, 32, 34, 36, 38, ... A005843 Composite numbers ..., 27, 28, 30, 32, 33, 34, 35, ... A002808 Powers of 2 ..., 4, 8, 16, 32, 64, 128, 256, ... A000079

## Sequences pertaining to 32

 Multiples of 32 0, 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, ... A174312 $3x+1$ sequence beginning at 21 21, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... A033481

## Partitions of 32

There are 8349 partitions of 32. Of these, several partitions do the of the [FINISH WRITING]

## Roots and powers of 32

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

 ${\sqrt {32}}$ 5.65685424 A010487 32 2 1024 ${\sqrt[{3}]{32}}$ 3.17480210 A010603 32 3 32768 ${\sqrt[{4}]{32}}$ 2.37841423 A011027 32 4 1048576 ${\sqrt[{5}]{32}}$ 2.00000000 32 5 33554432 ${\sqrt[{6}]{32}}$ 1.78179743 32 6 1073741824 ${\sqrt[{7}]{32}}$ 1.64067071 32 7 34359738368 ${\sqrt[{8}]{32}}$ 1.54221082 32 8 1099511627776 ${\sqrt[{9}]{32}}$ 1.46973449 32 9 35184372088832 ${\sqrt[{10}]{32}}$ 1.41421356 A002193 32 10 1125899906842624 A009976

## Logarithms and 32nd 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.

 $\log _{32}2$ 0.2 $\log _{2}32$ A102525 2 32 4294967296 $\log _{32}e$ 0.288539 A131920* $\log 32$ 3.46573590 A016655 $e^{32}$ 7.89629601... × 10 13 $\log _{32}3$ 0.316993 A020861* $\log _{3}32$ 3.15464876 A152747** 3 32 1853020188851841 $\log _{32}\pi$ 3.02756 $\log _{\pi }32$ 0.33029922 $\pi ^{32}$ 8.10580078... × 10 15 $\log _{32}4$ 0.4 $\log _{4}32$ 2.50000000 4 32 18446744073709551616 $\log _{32}5$ 0.464386 $\log _{5}32$ 2.15338279 A152675 5 32 23283064365386962890625 $\log _{32}6$ 0.516992 $\log _{6}32$ 1.93426403 6 32 7958661109946400884391936 $\log _{32}7$ 0.561471 $\log _{7}32$ 1.78103593 7 32 1104427674243920646305299201 $\log _{32}8$ 0.6 $\log _{8}32$ 1.66666666 A020793 8 32 79228162514264337593543950336 $\log _{32}9$ 0.633985 $\log _{9}32$ 1.57732438 9 32 3433683820292512484657849089281 $\log _{32}10$ 0.664386 $\log _{10}32$ 1.50514997 10 32 100000000000000000000000000000000

* Divided by 10.

** Multiplied by 10.

(See A001016 for the 32nd powers of integers).

## Values for number theoretic functions with 32 as an argument

 $\mu (32)$ 0 $M(32)$ –4 $\pi (32)$ 11 $\sigma _{1}(32)$ 63 $\sigma _{0}(32)$ 6 $\phi (32)$ 16 $\Omega (32)$ 5 $\omega (32)$ 1 $\lambda (32)$ 8 This is the Carmichael lambda function. $\lambda (32)$ –1 This is the Liouville lambda function. $\zeta (32)$ 32! 263130836933693530167218012160000000 $\Gamma (32)$ 8222838654177922817725562880000000

## Factorization of 32 in some quadratic integer rings

In $\mathbb {Z}$ , 32 has the prime factorization of 2 5. But it has different factorizations in some quadratic integer rings.

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

## Representation of 32 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 100000 1012 200 112 52 44 40 35 32 2A 28 26 24 22 20 1F 1E 1D 1C

REMARKS GO HERE