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# 16

Please do not rely on any information it contains.

16 is the square of 4 and the fourth power of 2. And it is the only integer satisfying ${\displaystyle x^{y}=y^{x}}$ with ${\displaystyle x\neq y}$.

## Membership in core sequences

 Even numbers ..., 10, 12, 14, 16, 18, 20, 22, ... A005843 Composite numbers ..., 12, 14, 15, 16, 18, 20, 21, ... A002808 Perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, ... A000290 Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256, ... A000079 Powers of 4 1, 4, 16, 64, 256, 1024, 4096, ... A000302

## Sequences pertaining to 16

 Multiples of 16 0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, ... A008598 Hexadecagonal numbers 1, 16, 45, 88, 145, 216, 301, 400, 513, 640, 781, 936, ... A051868 Pseudoprimes to base 16 15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, ... A020144 ${\displaystyle 3x+1}$ sequence starting at 3 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ... A033478

Assuming the Collatz conjecture is true, 8 and 16 appear in the Collatz trajectory of every positive integer except 1, 2, 4.

## Partitions of 16

There are 231 partitions of 16.

But there are only two Goldbach representations of 16: 3 + 13 = 5 + 11.

## Roots and powers of 16

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

 ${\displaystyle {\sqrt {16}}}$ 4.00000000 16 2 256 ${\displaystyle {\sqrt[{3}]{16}}}$ 2.51984209 A010588 16 3 4096 ${\displaystyle {\sqrt[{4}]{16}}}$ 2.00000000 16 4 65536 ${\displaystyle {\sqrt[{5}]{16}}}$ 1.74110112 A011101 16 5 1048576 ${\displaystyle {\sqrt[{6}]{16}}}$ 1.58740105 A005480 16 6 16777216 ${\displaystyle {\sqrt[{7}]{16}}}$ 1.48599428 A011366 16 7 268435456 ${\displaystyle {\sqrt[{8}]{16}}}$ 1.41421356 A002193 16 8 4294967296 ${\displaystyle {\sqrt[{9}]{16}}}$ 1.36079000 A011368 16 9 68719476736 ${\displaystyle {\sqrt[{10}]{16}}}$ 1.31950791 A005533 16 10 1099511627776 ${\displaystyle {\sqrt[{11}]{16}}}$ 1.28666489 A011370 16 11 17592186044416 ${\displaystyle {\sqrt[{12}]{16}}}$ 1.25992104 A002580 16 12 281474976710656 ${\displaystyle {\sqrt[{13}]{16}}}$ 1.23772628 A011372 16 13 4503599627370496 ${\displaystyle {\sqrt[{14}]{16}}}$ 1.21901365 A011186 16 14 72057594037927936 ${\displaystyle {\sqrt[{15}]{16}}}$ 1.20302503 A011374 16 15 1152921504606846976 ${\displaystyle {\sqrt[{16}]{16}}}$ 1.18920711 A010767 16 16 18446744073709551616 A001025

Of course the roots given above are the principal real roots. There are also negative real roots and complex roots.

• 4, –4 (both real)
• ${\displaystyle {\sqrt[{3}]{16}}}$, ${\displaystyle -{\sqrt[{3}]{2}}\pm {\sqrt[{3}]{2}}{\sqrt {-3}}}$ (the two complex roots are the same except for the sign of the imaginary part, and of course we're using the radical sign to refer to principal roots)
• 2, –2, ${\displaystyle 2i}$, ${\displaystyle -2i}$
• ${\displaystyle {\sqrt[{5}]{16}}}$, etc., and so on and so forth.

## Logarithms and sixteenth 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.

From Fermat's little theorem it follows that if ${\displaystyle b}$ is coprime to 17, then ${\displaystyle b^{16}\equiv 1\mod 17}$.

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

 ${\displaystyle \log _{16}2}$ 0.25 A020773 ${\displaystyle \log _{2}16}$ 4 2 16 65536 ${\displaystyle \log _{16}e}$ 0.360674 ${\displaystyle \log 16}$ 2.77259 A016639 ${\displaystyle e^{16}}$ 8.88611e+06 ${\displaystyle \log _{16}3}$ 0.396241 A153019 ${\displaystyle \log _{3}16}$ 2.52372 A154751 3 16 4.30467e+07 ${\displaystyle \log _{16}\pi }$ 0.412874 ${\displaystyle \log _{\pi }16}$ 2.42205 ${\displaystyle \pi ^{16}}$ 9.00322e+07 ${\displaystyle \log _{16}4}$ 0.5 A020761 ${\displaystyle \log _{4}16}$ 2 4 16 4.29497e+09 ${\displaystyle \log _{16}5}$ 0.580482 A153420 ${\displaystyle \log _{5}16}$ 1.72271 A154759 5 16 1.52588e+11 ${\displaystyle \log _{16}6}$ 0.646241 A153606 ${\displaystyle \log _{6}16}$ 1.54741 A154776 6 16 2.82111e+12 ${\displaystyle \log _{16}7}$ 0.701839 A153626 ${\displaystyle \log _{7}16}$ 1.42483 A154793 7 16 3.32329e+13 ${\displaystyle \log _{16}8}$ 0.75 A152627 ${\displaystyle \log _{8}16}$ 1.33333 A122553 8 16 2.81475e+14 ${\displaystyle \log _{16}9}$ 0.792481 A094148 ${\displaystyle \log _{9}16}$ 1.26186 A100831 9 16 1.85302e+15 ${\displaystyle \log _{16}10}$ 0.830482 A154166 ${\displaystyle \log _{10}16}$ 1.20412 A154794 10 16 1e+16

(See A010804 for the sixteenth powers of integers).

## Values for number theoretic functions with 16 as an argument

 ${\displaystyle \mu (16)}$ 0 ${\displaystyle M(16)}$ –1 ${\displaystyle \pi (16)}$ 6 ${\displaystyle \sigma _{1}(16)}$ 31 ${\displaystyle \sigma _{0}(16)}$ 5 ${\displaystyle \phi (16)}$ 8 ${\displaystyle \Omega (16)}$ 4 ${\displaystyle \omega (16)}$ 1 ${\displaystyle \lambda (16)}$ 4 This is the Carmichael lambda function. ${\displaystyle \lambda (16)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (16)}$ 1.0000152822594... ${\displaystyle {\frac {3617\pi ^{16}}{325641566250}}}$. See A013674. 16! 20922789888000 ${\displaystyle \Gamma (16)}$ 1307674368000

## Factorization of 16 in some quadratic integer rings

As was mentioned above, 16 is the fourth power of 2 in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings. We could just take the factorizations of 2 and stick in a bunch of 4s as exponents, change some exponent 2s to exponent 8s. That works, at least for those rings that are UFDs, but it seems to ignore rings that are not UFDs.

And yet, it is possible for an integer that is composite in ${\displaystyle \mathbb {Z} }$ to still have only one factorization in a non-UFD. In the case of 16, its only factorization in most imaginary quadratic integer rings is 2 4.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1-i)^{4}(1+i)^{4}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle ({\sqrt {-2}})^{8}}$. ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{8}}$ ${\displaystyle \mathbb {Z} [\omega ]}$ 2 4 ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (1-{\sqrt {3}})^{4}(1+{\sqrt {3}})^{4}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle \mathbb {Z} [\phi ]}$ 2 4 ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2-{\sqrt {6}})^{4}(2+{\sqrt {6}})^{4}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)^{4}\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)^{4}}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3-{\sqrt {7}})^{4}(3+{\sqrt {7}})^{4}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 4 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 4 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3-{\sqrt {11}})^{4}(3+{\sqrt {11}})^{4}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 4 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (4-{\sqrt {14}})^{4}(4+{\sqrt {14}})^{4}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 4 ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle \left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)^{4}\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)^{4}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (13-3{\sqrt {19}})^{4}(13+3{\sqrt {19}})^{4}}$

Note that there isn't a −1 in the factorization of 2 in ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$, because ${\displaystyle ({\sqrt {-2}})^{4n}=4^{n}}$.

## Representation of 16 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 through 36 Representation 10000 121 100 31 24 22 20 17 16 15 14 13 12 11 10 G

16 is a Harshad number in every base from binary to hexadecimal except 6, 10, 11, 12, 14.

 ${\displaystyle -1}$ 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