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# 31

Please do not rely on any information it contains.

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 ${\displaystyle 2^{n}-1}$ 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 ${\displaystyle 3x+1}$ sequence starting at 27 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, ... A008884 ${\displaystyle 3x-1}$ 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.

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

## Logarithms and 31st 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.

 ${\displaystyle \log _{31}2}$ 0.201849 ${\displaystyle \log _{2}31}$ 4.9542 2 31 2147483648 ${\displaystyle \log _{31}e}$ 0.291207 ${\displaystyle \log 31}$ 3.43399 A016654 ${\displaystyle e^{31}}$ ${\displaystyle \log _{31}3}$ 0.319923 ${\displaystyle \log _{3}31}$ 3.12575 3 31 617673396283947 ${\displaystyle \log _{31}4}$ 0.403698 ${\displaystyle \log _{4}31}$ 2.4771 4 31 4611686018427387904 ${\displaystyle \log _{31}5}$ 0.468679 ${\displaystyle \log _{5}31}$ 2.13366 5 31 4656612873077392578125 ${\displaystyle \log _{31}6}$ 0.521772 ${\displaystyle \log _{6}31}$ 1.91654 6 31 1326443518324400147398656 ${\displaystyle \log _{31}7}$ 0.566662 ${\displaystyle \log _{7}31}$ 1.76472 7 31 157775382034845806615042743 ${\displaystyle \log _{31}8}$ 0.605547 ${\displaystyle \log _{8}31}$ 1.6514 8 31 9903520314283042199192993792 ${\displaystyle \log _{31}9}$ 0.639846 ${\displaystyle \log _{9}31}$ 1.56287 9 31 381520424476945831628649898809 ${\displaystyle \log _{31}10}$ 0.670528 ${\displaystyle \log _{10}31}$ 1.49136 10 31 10000000000000000000000000000000

## Values for number theoretic functions with 31 as an argument

 ${\displaystyle \mu (31)}$ –1 ${\displaystyle M(31)}$ –4 ${\displaystyle \pi (31)}$ 11 ${\displaystyle \sigma _{1}(31)}$ 32 ${\displaystyle \sigma _{0}(31)}$ 2 ${\displaystyle \phi (31)}$ 30 ${\displaystyle \Omega (31)}$ 1 ${\displaystyle \omega (31)}$ 1 ${\displaystyle \lambda (31)}$ 30 This is the Carmichael lambda function. ${\displaystyle \lambda (31)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (31)}$ 31! 8222838654177922817725562880000000 ${\displaystyle \Gamma (31)}$ 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 ${\displaystyle \mathbb {Z} [{\sqrt {31}}]}$, with units of the form ${\displaystyle \pm (1520+273{\sqrt {31}})^{n}\,}$ (${\displaystyle n\in \mathbb {Z} }$), is a unique factorization domain, but it is not norm-Euclidean. ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-31}})}}$ is not Euclidean for any function whatsoever, nor is it a UFD at all, having class number 3.

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

## Factorization of 31 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 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

 ${\displaystyle -1}$ 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