11

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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
$3x+1$ sequence beginning at 9	9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ...	A033478
$5x+1$ 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.

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

Logarithms and eleventh 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.

If $n$ is not a multiple of 23, then either $n^{11}-1$ or $n^{11}+1$ is. Hence the formula for the Legendre symbol $\left(\frac{a}{23}\right)=a^{11}mod23$.

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

${\log }_{11}2$	0.28906482	A152748	${\log }_{2}11$	3.45943161	A020863	2 11	2048
${\log }_{11}e$	0.41703239		$\log 11$	2.39789527	A016634	$e^{11}$	59874.14171519
${\log }_{11}3$	0.45815690	A152974	${\log }_{3}11$	2.18265833	A154175	3 11	177147
${\log }_{11}\pi$	0.47738944		${\log }_{\pi }11$	2.09472584		${\pi }^{11}$	294204.01797389
${\log }_{11}4$	0.57812965	A153104	${\log }_{4}11$	1.72971580	A154176	4 11	4194304
${\log }_{11}5$	0.67118774	A153269	${\log }_{5}11$	1.48989610	A154177	5 11	48828125
${\log }_{11}6$	0.74722173	A153586	${\log }_{6}11$	1.33829083	A154178	6 11	362797056
${\log }_{11}7$	0.81150756	A153621	${\log }_{7}11$	1.23227440	A154179	7 11	1977326743
${\log }_{11}8$	0.86719447	A153791	${\log }_{8}11$	1.15314387	A154180	8 11	8589934592
${\log }_{11}9$	0.91631381	A154011	${\log }_{9}11$	1.09132916	A154181	9 11	31381059609
${\log }_{11}10$	0.96025256	A154161	${\log }_{10}11$	1.04139268	A154182	10 11	100000000000

Values for number theoretic functions with 11 as an argument

$\mu (11)$	–1
$M(11)$	–2
$\pi (11)$	5
${\sigma }_1(11)$	12
${\sigma }_0(11)$	2
$\varphi (11)$	10
$\mathrm{\Omega }(11)$	1
$\omega (11)$	1
$\lambda (11)$	10	This is the Carmichael lambda function.
$\lambda (11)$	–1	This is the Liouville lambda function.
$\zeta (11)$	1.0004941886041194645587... (see A013669).
11!	39916800
$\mathrm{\Gamma }(11)$	3628800

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

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

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

Factorization of 11 in some quadratic integer rings

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

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

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