16

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

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
$3x+1$ 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.

${\sqrt {16}}$	4.00000000		16 2	256
${\sqrt[{3}]{16}}$	2.51984209	A010588	16 3	4096
${\sqrt[{4}]{16}}$	2.00000000		16 4	65536
${\sqrt[{5}]{16}}$	1.74110112	A011101	16 5	1048576
${\sqrt[{6}]{16}}$	1.58740105	A005480	16 6	16777216
${\sqrt[{7}]{16}}$	1.48599428	A011366	16 7	268435456
${\sqrt[{8}]{16}}$	1.41421356	A002193	16 8	4294967296
${\sqrt[{9}]{16}}$	1.36079000	A011368	16 9	68719476736
${\sqrt[{10}]{16}}$	1.31950791	A005533	16 10	1099511627776
${\sqrt[{11}]{16}}$	1.28666489	A011370	16 11	17592186044416
${\sqrt[{12}]{16}}$	1.25992104	A002580	16 12	281474976710656
${\sqrt[{13}]{16}}$	1.23772628	A011372	16 13	4503599627370496
${\sqrt[{14}]{16}}$	1.21901365	A011186	16 14	72057594037927936
${\sqrt[{15}]{16}}$	1.20302503	A011374	16 15	1152921504606846976
${\sqrt[{16}]{16}}$	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)
• ${\sqrt[{3}]{16}}$, $-{\sqrt[{3}]{2}}\pm {\sqrt[{3}]{2}}{\sqrt {-3}}$ (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, $2i$, $-2i$
• ${\sqrt[{5}]{16}}$, etc., and so on and so forth.

Logarithms and sixteenth 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.

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

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

$\log _{16}2$	0.25000000	A020773	$\log _{2}16$	4.00000000		2 16	65536
$\log _{16}e$	0.36067376		$\log 16$	2.77258872	A016639	$e^{16}$	8886110.52050787
$\log _{16}3$	0.39624062	A153019	$\log _{3}16$	2.52371901	A154751	3 16	43046721
$\log _{16}\pi$	0.41287403		$\log _{\pi }16$	2.42204624		$\pi ^{16}$	90032220.84293327
$\log _{16}4$	0.50000000	A020761	$\log _{4}16$	2.00000000		4 16	4294967296
$\log _{16}5$	0.58048202	A153420	$\log _{5}16$	1.72270623	A154759	5 16	152587890625
$\log _{16}6$	0.64624062	A153606	$\log _{6}16$	1.54741122	A154776	6 16	2821109907456
$\log _{16}7$	0.70183873	A153626	$\log _{7}16$	1.42482874	A154793	7 16	33232930569601
$\log _{16}8$	0.75000000	A152627	$\log _{8}16$	1.33333333	A122553	8 16	281474976710656
$\log _{16}9$	0.79248125	A094148	$\log _{9}16$	1.26185950	A100831	9 16	1853020188851841
$\log _{16}10$	0.83048202	A154166	$\log _{10}16$	1.20411998	A154794	10 16	10000000000000000

(See A010804 for the sixteenth powers of integers).

Values for number theoretic functions with 16 as an argument

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

Factorization of 16 in some quadratic integer rings

As was mentioned above, 16 is the fourth power of 2 in $\mathbb {Z}$. 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 $\mathbb {Z}$ 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.

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

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

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.

See also

• 2^n mod n

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