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# 11

Please do not rely on any information it contains.

11 is an integer with the largest known multiplicative persistence in base 10 (A003001, A031346).

## Membership in core sequences

 Odd numbers ..., 5, 7, 9, 11, 13, 15, 17, 19, 21, ... A005408 Prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... A000040 Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... A000032 Jacobsthal numbers 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... A001045

In Pascal's triangle, 11 occurs twice. (In Lozanić's triangle, 11 occurs four times).

## Sequences pertaining to 11

 Multiples of 11 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, ... A008593 Fermat pseudoprimes to base 11 10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, ... A020139 ${\displaystyle 3x+1}$ sequence beginning at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ... A033478 ${\displaystyle 5x+1}$ sequence beginning at 11 11, 56, 28, 14, 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, ... A259193 11-rough numbers 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ... A008364

## Partitions of 11

There are 56 partitions of 11.

## Roots and powers of 11

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

 ${\displaystyle {\sqrt {11}}}$ 3.31662479 A010468 11 2 121 ${\displaystyle {\sqrt[{3}]{11}}}$ 2.22398009 A010583 11 3 1331 ${\displaystyle {\sqrt[{4}]{11}}}$ 1.82116028 A011008 11 4 14641 ${\displaystyle {\sqrt[{5}]{11}}}$ 1.61539426 A011096 11 5 161051 ${\displaystyle {\sqrt[{6}]{11}}}$ 1.49130147 A011290 11 6 1771561 ${\displaystyle {\sqrt[{7}]{11}}}$ 1.40854388 A011291 11 7 19487171 ${\displaystyle {\sqrt[{8}]{11}}}$ 1.34950371 A011292 11 8 214358881 ${\displaystyle {\sqrt[{9}]{11}}}$ 1.30529988 A011293 11 9 2357947691 ${\displaystyle {\sqrt[{10}]{11}}}$ 1.27098161 A011294 11 10 25937424601 ${\displaystyle {\sqrt[{11}]{11}}}$ 1.24357522 A011295 11 11 285311670611 A001020

## Logarithms and eleventh 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.

If ${\displaystyle n}$ is not a multiple of 23, then either ${\displaystyle n^{11}-1}$ or ${\displaystyle n^{11}+1}$ is. Hence the formula for the Legendre symbol ${\displaystyle \left({\frac {a}{23}}\right)=a^{11}\mod 23}$.

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

 ${\displaystyle \log _{11}2}$ 0.289065 A152748 ${\displaystyle \log _{2}11}$ 3.45943 A020863 2 11 2048 ${\displaystyle \log _{11}e}$ 0.417032 ${\displaystyle \log 11}$ 2.3979 A016634 ${\displaystyle e^{11}}$ 59874.1 ${\displaystyle \log _{11}3}$ 0.458157 A152974 ${\displaystyle \log _{3}11}$ 2.18266 A154175 3 11 177147 ${\displaystyle \log _{11}\pi }$ 0.477389 ${\displaystyle \log _{\pi }11}$ 2.09473 ${\displaystyle \pi ^{11}}$ 294204 ${\displaystyle \log _{11}4}$ 0.57813 A153104 ${\displaystyle \log _{4}11}$ 1.72972 A154176 4 11 4.1943e+06 ${\displaystyle \log _{11}5}$ 0.671188 A153269 ${\displaystyle \log _{5}11}$ 1.4899 A154177 5 11 4.88281e+07 ${\displaystyle \log _{11}6}$ 0.747222 A153586 ${\displaystyle \log _{6}11}$ 1.33829 A154178 6 11 3.62797e+08 ${\displaystyle \log _{11}7}$ 0.811508 A153621 ${\displaystyle \log _{7}11}$ 1.23227 A154179 7 11 1.97733e+09 ${\displaystyle \log _{11}8}$ 0.867194 A153791 ${\displaystyle \log _{8}11}$ 1.15314 A154180 8 11 8.58993e+09 ${\displaystyle \log _{11}9}$ 0.916314 A154011 ${\displaystyle \log _{9}11}$ 1.09133 A154181 9 11 3.13811e+10 ${\displaystyle \log _{11}10}$ 0.960253 A154161 ${\displaystyle \log _{10}11}$ 1.04139 A154182 10 11 1e+11

## Values for number theoretic functions with 11 as an argument

 ${\displaystyle \mu (11)}$ –1 ${\displaystyle M(11)}$ –2 ${\displaystyle \pi (11)}$ 5 ${\displaystyle \sigma _{1}(11)}$ 12 ${\displaystyle \sigma _{0}(11)}$ 2 ${\displaystyle \phi (11)}$ 10 ${\displaystyle \Omega (11)}$ 1 ${\displaystyle \omega (11)}$ 1 ${\displaystyle \lambda (11)}$ 10 This is the Carmichael lambda function. ${\displaystyle \lambda (11)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (11)}$ 1.0004941886041194645587... (see A013669). 11! 39916800 ${\displaystyle \Gamma (11)}$ 3628800

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −11, 11

Both ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ and ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ are unique factorization domains.

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

## Factorization of 11 in some quadratic integer rings

As was mentioned above, 11 is a prime number in ${\displaystyle \mathbb {Z} }$. But it is composite in some quadratic integer rings. As was mentioned above, 11 is a prime number in ${\displaystyle \mathbb {Z} }$. But it is composite in some quadratic integer rings.

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

## Representation of 11 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 through 36 Representation 1011 102 23 21 15 14 13 12 11 10 B

Clearly 11 is a palindromic number in base 10. However, and this may seem rather counter-intuitive, it is also a strictly non-palindromic number (A016038). As the chart above shows, it is not palindromic in binary, nor any other base up to base 9.

 ${\displaystyle -1}$ 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