31

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31 is an integer.

Membership in core sequences

Odd numbers	..., 25, 27, 29, 31, 33, 35, 37, ...	A005408
Prime numbers	..., 19, 23, 29, 31, 37, 41, 43, ...	A000040
Squarefree numbers	..., 26, 29, 30, 31, 33, 34, 35, ...	A005117
Numbers of the form $2^n-1$	1, 3, 7, 15, 31, 63, 127, 255, 511, ...	A000225
Mersenne exponents	2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...	A000043

Note that in addition to being the exponent of a Mersenne prime (2147483647), 31 is itself a Mersenne prime, with its corresponding exponent being 5.

In Pascal's triangle, 31 occurs twice.

Sequences pertaining to 31

Multiples of 31	0, 31, 62, 93, 124, 155, 186, 217, 248, 279, ...	A135631
Primes with primitive root 31	2, 7, 17, 29, 47, 53, 59, 61, 67, 71, 73, 89, ...	A019357
$3x+1$ sequence starting at 27	27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, ...	A008884
$3x-1$ sequence starting at 84	84, 42, 21, 62, 31, 92, 46, 23, 68, 34, 17, 50, ...	A008898

Partitions of 31

There are 6842 partitions of 31.

The Goldbach representations of 31 using distinct primes are: 2 + 29 = 3 + 5 + 23 = 3 + 11 + 17 = 5 + 7 + 19 = 7 + 11 + 13 = 31.

Roots and powers of 31

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

$\sqrt{31}$	5.56776436	A010486	31 2	961
$\sqrt[3]{31}$	3.14138065	A010602	31 3	29791
$\sqrt[4]{31}$	2.35961106	A011026	31 4	923521
$\sqrt[5]{31}$	1.98734075	A011116	31 5	28629151
$\sqrt[6]{31}$	1.77239404		31 6	887503681
$\sqrt[7]{31}$	1.63324625		31 7	27512614111
$\sqrt[8]{31}$	1.53610255		31 8	852891037441
$\sqrt[9]{31}$	1.46455894		31 9	26439622160671
$\sqrt[10]{31}$	1.40973073		31 10	819628286980801
				A009975

Logarithms and 31st 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.

${\log }_{31}2$	0.20184908		${\log }_{2}31$	4.95419631		2 31	2147483648
${\log }_{31}e$	0.29120667		$\log 31$	3.43398720	A016654	$e^{31}$
${\log }_{31}3$	0.31992323		${\log }_{3}31$	3.12574985		3 31	617673396283947
${\log }_{31}4$	0.40369817		${\log }_{4}31$	2.47709815		4 31	4611686018427387904
${\log }_{31}5$	0.46867906		${\log }_{5}31$	2.13365621		5 31	4656612873077392578125
${\log }_{31}6$	0.52177231		${\log }_{6}31$	1.91654475		6 31	1326443518324400147398656
${\log }_{31}7$	0.56666202		${\log }_{7}31$	1.76472033		7 31	157775382034845806615042743
${\log }_{31}8$	0.60554725		${\log }_{8}31$	1.65139877		8 31	9903520314283042199192993792
${\log }_{31}9$	0.63984646		${\log }_{9}31$	1.56287492		9 31	381520424476945831628649898809
${\log }_{31}10$	0.67052815		${\log }_{10}31$	1.49136169		10 31	10000000000000000000000000000000

Values for number theoretic functions with 31 as an argument

$\mu (31)$	–1
$M(31)$	–4
$\pi (31)$	11
${\sigma }_1(31)$	32
${\sigma }_0(31)$	2
$\varphi (31)$	30
$\mathrm{\Omega }(31)$	1
$\omega (31)$	1
$\lambda (31)$	30	This is the Carmichael lambda function.
$\lambda (31)$	–1	This is the Liouville lambda function.
$\zeta (31)$
31!	8222838654177922817725562880000000
$\mathrm{\Gamma }(31)$	265252859812191058636308480000000

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −31, 31

The commutative quadratic integer ring with unity $\mathbb{\mathbb{Z}}[\sqrt{31}]$, with units of the form $\pm (1520+273\sqrt{31}{)}^n$ ($n\in \mathbb{\mathbb{Z}}$), is a unique factorization domain, but it is not norm-Euclidean. ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-31})}$ is not Euclidean for any function whatsoever, nor is it a UFD at all, having class number 3.

$n$	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-31})}$	$\mathbb{\mathbb{Z}}[\sqrt{31}]$
2	Irreducible	$(39-7\sqrt{31})(39+7\sqrt{31})$
3		$(-1)(11-2\sqrt{31})(11+2\sqrt{31})$
4	2 2	$(39\pm 7\sqrt{31}{)}^2$
5	Irreducible	$(6-\sqrt{31})(6+\sqrt{31})$
6	2 × 3	$(-1)(39\pm 7\sqrt{31})(11\pm 2\sqrt{31})$
7	Irreducible	Prime
8	2 3 OR $(\frac{1}{2}-\frac{\sqrt{-31}}{2})(\frac{1}{2}+\frac{\sqrt{-31}}{2})$	$(39\pm 7\sqrt{31}{)}^3$
9	3 2	$(11\pm 2\sqrt{31}{)}^2$
10	2 × 5 OR $(\frac{3}{2}-\frac{\sqrt{-31}}{2})(\frac{3}{2}+\frac{\sqrt{-31}}{2})$	$(39\pm 7\sqrt{31})(6\pm \sqrt{31})$
11	Irreducible	$(-1)(50-9\sqrt{31})(50+9\sqrt{31})$
12	2 2 × 3	$(-1)(39\pm 7\sqrt{31}{)}^2(11\pm 2\sqrt{31})$
13	Irreducible	Prime
14	2 × 7 OR $(\frac{5}{2}-\frac{\sqrt{-31}}{2})(\frac{5}{2}+\frac{\sqrt{-31}}{2})$	$(39\pm 7\sqrt{31})7$
15	3 × 5	$(-1)(11\pm 2\sqrt{31})(6\pm \sqrt{31})$
16	2 4	$(39\pm 7\sqrt{31}{)}^4$
17	Irreducible	Prime
18	2 × 3 2	$(39\pm 7\sqrt{31})(11\pm 2\sqrt{31}{)}^2$
19	Irreducible	Prime
20	2 2 × 5 OR $2(\frac{3}{2}-\frac{\sqrt{-31}}{2})(\frac{3}{2}+\frac{\sqrt{-31}}{2})$ OR $(\frac{7}{2}-\frac{\sqrt{-31}}{2})(\frac{7}{2}+\frac{\sqrt{-31}}{2})$	$(39\pm 7\sqrt{31}{)}^2(6\pm \sqrt{31})$

Factorization of 31 in some quadratic integer rings

In $\mathbb{\mathbb{Z}}$, 31 is a prime number. But in some quadratic integer rings, it is composite.

TABLE GOES HERE

Representation of 31 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	11111	1011	133	111	51	43	37	34	31	29	27	25	23	21	1F	1E	1D	1C	1B

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