29

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29 is an integer.

Membership in core sequences

Odd numbers	..., 23, 25, 27, 29, 31, 33, 35, ...	A005408
Prime numbers	..., 17, 19, 23, 29, 31, 37, 41, ...	A000040
Squarefree numbers	..., 22, 23, 26, 29, 30, 31, 33, ...	A005117
Lucas numbers	..., 7, 11, 18, 29, 47, 76, 123, ...	A000032
Pell numbers	..., 2, 5, 12, 29, 70, 169, 408, ...	A000129
Central polygonal numbers	..., 11, 16, 22, 29, 37, 46, 56, ...	A000124

In Pascal's triangle, 29 occurs twice. (In Lozanić's triangle, 29 occurs four times).

Sequences pertaining to 29

Multiples of 29	0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, ...	A195819
29-gonal numbers	1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794, ...	A255187
29-gonal pyramidal numbers	1, 30, 114, 280, 555, 966, 1540, 2304, 3285, 4510, 6006, ...	A256649
Primes with primitive root 29	2, 3, 11, 17, 19, 41, 43, 47, 73, 79, 89, 97, 101, 113, 127, ...	A019355
$3x+1$ sequence starting at 51	51, 154, 77, 232, 116, 58, 29, 88, 44, 22, 11, 34, 17, 52, ...	A033479
$5x+1$ sequence starting at 7	7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, ...	A028389

Partitions of 29

There are 4565 partitions of 29.

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

Roots and powers of 29

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

$\sqrt{29}$	5.38516480	A010484	29 2	841
$\sqrt[3]{29}$	3.07231682	A010600	29 3	24389
$\sqrt[4]{29}$	2.32059578	A011024	29 4	707281
$\sqrt[5]{29}$	1.96100905	A011114	29 5	20511149
$\sqrt[6]{29}$	1.75280256		29 6	594823321
$\sqrt[7]{29}$	1.61775965		29 7	17249876309
$\sqrt[8]{29}$	1.52335018		29 8	500246412961
$\sqrt[9]{29}$	1.45374644		29 9	14507145975869
$\sqrt[10]{29}$	1.40036033		29 10	420707233300201
				A009973

Logarithms and 29th 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 59, then either $n^{29}-1$ or $n^{29}+1$ is. Hence the formula for the Legendre symbol $\left(\frac{a}{59}\right)=a^{29}(\mathrm{mod}59)$.

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

${\log }_{29}2$	0.20584683		${\log }_{2}29$	4.85798099		2 29	536870912
${\log }_{29}e$	0.29697420		$\log 29$	3.36729582	A016652	$e^{29}$	3931334297144.04207438
${\log }_{29}3$	0.32625951		${\log }_{3}29$	3.06504475		3 29	68630377364883
${\log }_{29}4$	0.41169366		${\log }_{4}29$	2.42899049		4 29	288230376151711744
${\log }_{29}5$	0.47796154		${\log }_{5}29$	2.09221853		5 29	186264514923095703125
${\log }_{29}6$	0.53210634		${\log }_{6}29$	1.87932358		6 29	36845653286788892983296
${\log }_{29}7$	0.57788511		${\log }_{7}29$	1.73044774		7 29	3219905755813179726837607
${\log }_{29}8$	0.61754049		${\log }_{8}29$	1.61932699		8 29	154742504910672534362390528
${\log }_{29}9$	0.65251902		${\log }_{9}29$	1.53252237		9 29	4710128697246244834921603689
${\log }_{29}10$	0.68380837		${\log }_{10}29$	1.46239799		10 29	100000000000000000000000000000

See A122970 for the 29th powers of integers.

Values for number theoretic functions with 29 as an argument

$\mu (29)$	−1
$M(29)$	−2
$\pi (29)$	10
${\sigma }_1(29)$	30
${\sigma }_0(29)$	2
$\varphi (29)$	28
$\mathrm{\Omega }(29)$	1
$\omega (29)$	1
$\lambda (29)$	28	This is the Carmichael lambda function.
$\lambda (29)$	−1	This is the Liouville lambda function.
29!	8841761993739701954543616000000
$\mathrm{\Gamma }(29)$	304888344611713860501504000000

Factorization of some small integers in a quadratic integer ring adjoining square roots of −29, 29

${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{29})}$ is a unique factorization domain, $\mathbb{\mathbb{Z}}[\sqrt{-29}]$ is not. Units in ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{29})}$ are of the form ${(\frac{5}{2}+\frac{\sqrt{29}}{2})}^n$. Units in $\mathbb{\mathbb{Z}}[\sqrt{-29}]$ are just 1 and −1.

$n$	$\mathbb{\mathbb{Z}}[\sqrt{-29}]$	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{29})}$
2	Irreducible	Prime
3
4	2 2
5	Irreducible	$(-1)(\frac{3}{2}-\frac{\sqrt{29}}{2})(\frac{3}{2}+\frac{\sqrt{29}}{2})$
6	2 × 3
7	Prime	$(-1)(\frac{1}{2}-\frac{\sqrt{29}}{2})(\frac{1}{2}+\frac{\sqrt{29}}{2})$
8	2 3
9	3 2
10	2 × 5	$(-1)2(\frac{3}{2}-\frac{\sqrt{29}}{2})(\frac{3}{2}+\frac{\sqrt{29}}{2})$
11	Irreducible	Prime
12	2 2 × 3
13	Irreducible	$(\frac{9}{2}-\frac{\sqrt{29}}{2})(\frac{9}{2}+\frac{\sqrt{29}}{2})$
14	2 × 7	$(-1)2(\frac{1}{2}-\frac{\sqrt{29}}{2})(\frac{1}{2}+\frac{\sqrt{29}}{2})$
15	3 × 5	$(-1)3(\frac{3}{2}-\frac{\sqrt{29}}{2})(\frac{3}{2}+\frac{\sqrt{29}}{2})$
16	2 4
17	Prime
18	2 × 3 2
19	Irreducible	Prime
20	2 2 × 5	$(-1)2^2(\frac{3}{2}-\frac{\sqrt{29}}{2})(\frac{3}{2}+\frac{\sqrt{29}}{2})$
21	3 × 7	$(-1)3(\frac{1}{2}-\frac{\sqrt{29}}{2})(\frac{1}{2}+\frac{\sqrt{29}}{2})$
22	2 × 11
23	Prime	$(\frac{11}{2}-\frac{\sqrt{29}}{2})(\frac{11}{2}+\frac{\sqrt{29}}{2})$
24	2 3 × 3
25	5 2	${(\frac{3}{2}-\frac{\sqrt{29}}{2})}^2{(\frac{3}{2}+\frac{\sqrt{29}}{2})}^2$
26	2 × 13	$2(\frac{9}{2}\pm \frac{\sqrt{29}}{2})$
27	3 3
28	2 2 × 7	$(-1)2^2(\frac{1}{2}-\frac{\sqrt{29}}{2})(\frac{1}{2}+\frac{\sqrt{29}}{2})$
29	$(-1)(\sqrt{-29}{)}^2$	$(\sqrt{29}{)}^2$
30	2 × 3 × 5 OR $(1-\sqrt{-29})(1+\sqrt{-29})$	$(-1)2\times 3(\frac{3}{2}\pm \frac{\sqrt{29}}{2})$
31	Prime
32	2 5
33	3 × 11 OR $(2-\sqrt{-29})(2+\sqrt{-29})$	3 × 11
34	2 × 17
35	5 × 7	$(\frac{3}{2}\pm \frac{\sqrt{29}}{2})(\frac{1}{2}\pm \frac{\sqrt{29}}{2})$
36	2 2 × 3 2
37
38	2 × 19 OR $(3-\sqrt{-29})(3+\sqrt{-29})$	2 × 19
39	3 × 13	$3(\frac{9}{2}\pm \frac{\sqrt{29}}{2})$
40	2 3 × 5	$(-1)2^3(\frac{3}{2}-\frac{\sqrt{29}}{2})(\frac{3}{2}+\frac{\sqrt{29}}{2})$

$\mathbb{\mathbb{Z}}[\sqrt{-29}]$ has class number 6. Here we will exhibit a few more examples of numbers with more than one distinct factorization in $\mathbb{\mathbb{Z}}[\sqrt{-29}]$ in which the factorizations have differing numbers of irreducible factors.

$n$	$\mathbb{\mathbb{Z}}[\sqrt{-29}]$
45	3 2 × 5 OR $(4-\sqrt{-29})(4+\sqrt{-29})$
54	2 × 3 3 OR $(5-\sqrt{-29})(5+\sqrt{-29})$
78	2 × 3 × 13 OR $(7-\sqrt{-29})(7+\sqrt{-29})$
110	2 × 5 × 11 OR $(9-\sqrt{-29})(9+\sqrt{-29})$
117	3 2 × 13 OR $(1-2\sqrt{-29})(1+2\sqrt{-29})$
120	2 3 × 3 × 5 OR $(2-2\sqrt{-29})(2+2\sqrt{-29})$
125	5 3 OR $(3-2\sqrt{-29})(3+2\sqrt{-29})$

Ideals really help us make sense of multiple distinct factorizations in $\mathbb{\mathbb{Z}}[\sqrt{-29}]$, while raising some questions about ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{29})}$.

$p$	Factorization of $⟨p⟩$
	In $\mathbb{\mathbb{Z}}[\sqrt{-29}]$	In ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{29})}$
2	$⟨2,1+\sqrt{-29}{⟩}^2$	Prime
3	$⟨3,1-\sqrt{-29}⟩⟨3,1+\sqrt{-29}⟩$
5	$⟨5,1-\sqrt{-29}⟩⟨5,1+\sqrt{-29}⟩$
7	Prime
11	$⟨11,2-\sqrt{-29}⟩⟨11,2+\sqrt{-29}⟩$	Prime
13	$⟨13,6-\sqrt{-29}⟩⟨13,6+\sqrt{-29}⟩$	$⟨4-\sqrt{29}⟩⟨13,4+\sqrt{29}⟩$
17	Prime
19	$⟨19,3-\sqrt{29}⟩⟨19,3+\sqrt{29}⟩$	Prime
23	Prime
29	$⟨\sqrt{-29}{⟩}^2$	$⟨\sqrt{29}{⟩}^2$
31
37
41
43
47

Factorization of 29 in some quadratic integer rings

As was mentioned above, 29 is a prime number in $\mathbb{\mathbb{Z}}$. But it is composite in some quadratic integer rings.

$\mathbb{\mathbb{Z}}[i]$	$(2-5i)(2+5i)$
$\mathbb{\mathbb{Z}}[\sqrt{-2}]$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{2}]$	Prime
$\mathbb{\mathbb{Z}}[\omega ]$		$\mathbb{\mathbb{Z}}[\sqrt{3}]$
$\mathbb{\mathbb{Z}}[\sqrt{-5}]$	$(3-2\sqrt{-5})(3+2\sqrt{-5})$	$\mathbb{\mathbb{Z}}[\varphi ]$	$(-1)(20-13\varphi )(7+13\varphi )$
$\mathbb{\mathbb{Z}}[\sqrt{-6}]$	Irreducible	$\mathbb{\mathbb{Z}}[\sqrt{6}]$	$(-1)(11-5\sqrt{6})(11+5\sqrt{6})$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{7}]$	$(27-10\sqrt{7})(27+10\sqrt{7})$
$\mathbb{\mathbb{Z}}[\sqrt{-10}]$	Irreducible	$\mathbb{\mathbb{Z}}[\sqrt{10}]$	Irreducible
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-11})}$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{11}]$	Prime
$\mathbb{\mathbb{Z}}[\sqrt{-13}]$	Irreducible	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{13})}$	$(-1)(\frac{1}{2}-\frac{3\sqrt{13}}{2})(\frac{1}{2}+\frac{3\sqrt{13}}{2})$
$\mathbb{\mathbb{Z}}[\sqrt{-14}]$		$\mathbb{\mathbb{Z}}[\sqrt{14}]$	Prime
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-15})}$		$\mathbb{\mathbb{Z}}[\sqrt{15}]$
$\mathbb{\mathbb{Z}}[\sqrt{-17}]$		${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-19})}$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{19}]$

Representation of 29 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	11101	1002	131	104	45	41	35	32	29	27	25	23	21	1E	1D	1C	1B	1A	19

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