30

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30 is an integer. It is the largest integer such that all integers between 1 and itself coprime to it are prime (namely: 7, 11, 13, 17, 19, 23, 29; see A005776).

Membership in core sequences

Even numbers	..., 24, 26, 28, 30, 32, 34, 36, ...	A005843
Composite numbers	..., 26, 27, 28, 30, 32, 33, 34, ...	A002808
Squarefree numbers	..., 23, 26, 29, 30, 31, 33, 34, ...	A005117
Primorials	1, 2, 6, 30, 210, 2310, 30030, ...	A002110
Partition numbers	..., 11, 15, 22, 30, 42, 56, 77, ...	A000041

In Pascal's triangle, 30 occurs twice.

Sequences pertaining to 30

Multiples of 30	0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, ...	A249674
Divisors of 30	1, 2, 3, 5, 6, 10, 15, 30	A018255
Squares modulo 30	0, 1, 4, 6, 9, 10, 15, 16, 19, 21, 24, 25	A010462
Primes with primitive root 30	11, 23, 41, 43, 47, 59, 61, 79, 89, 109, 131, 151, 167, 173, ...	A019356
30-gonal numbers	1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860, ...	A254474
$3x+1$ sequence beginning at 30	30, 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ...

Partitions of 30

There are 5604 partitions of 30. Of these, the Goldbach representations are 23 + 7, 19 + 11 and 17 + 13.

There are four ways to represent 30 as a sum of distinct divisors (see A033630): 1 + 3 + 5 + 6 + 15 = 2 + 3 + 10 + 15 = 5 + 10 + 15 = 30.

Roots and powers of 30

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

$\sqrt{30}$	5.47722557	A010485	30 2	900
$\sqrt[3]{30}$	3.10723250	A010601	30 3	27000
$\sqrt[4]{30}$	2.34034731	A011025	30 4	810000
$\sqrt[5]{30}$	1.97435048	A011115	30 5	24300000
$\sqrt[6]{30}$	1.76273438		30 6	729000000
$\sqrt[7]{30}$	1.62561359		30 7	21870000000
$\sqrt[8]{30}$	1.52981937		30 8	656100000000
$\sqrt[9]{30}$	1.45923280		30 9	19683000000000
$\sqrt[10]{30}$	1.40511582		30 10	590490000000000
				A009974

Logarithms and thirtieth 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 61, then either $n^{30}-1$ or $n^{30}+1$ is. Hence the formula for the Legendre symbol $\left(\frac{a}{61}\right)=a^{30}(\mathrm{mod}61)$.

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

TABLE

Values for number theoretic functions with 30 as an argument

$\mu (30)$	−1
$M(30)$	−3
$\pi (30)$	8
${\sigma }_1(30)$	72
${\sigma }_0(30)$	8
$\varphi (30)$	8
$\mathrm{\Omega }(30)$	3
$\omega (30)$	3
$\lambda (30)$	4	This is the Carmichael lambda function.
$\lambda (30)$	−1	This is the Liouville lambda function.
30!	265252859812191058636308480000000
$\mathrm{\Gamma }(30)$	8841761993739701954543616000000

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −30, 30

The commutative quadratic integer ring with unity $\mathbb{\mathbb{Z}}[\sqrt{30}]$, with units of the form $\pm (11+2\sqrt{30}{)}^n$ ($n\in \mathbb{\mathbb{Z}}$), is not a unique factorization domain. But since 30 = 3 × 10, it follows that those primes having a least significant digit of 3 or 7 in base 10 are inert and irreducible in $\mathbb{\mathbb{Z}}[\sqrt{30}]$. But ending in 1 or 9 does not automatically guarantee the prime splits in $\mathbb{\mathbb{Z}}[\sqrt{30}]$.

$\mathbb{\mathbb{Z}}[\sqrt{-30}]$ is not a unique factorization domain either. However, its scarcity of units gives us greater confidence in identifying instances of non-unique factorization.

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

Quite surprisingly, $(\sqrt{30}{)}^2$ is a distinct factorization of 30. As shown in the table above, we can factorize 30 as $(-1)2\times 3(5-\sqrt{30})(5+\sqrt{30})$. Apart from the unit, dividing $\sqrt{30}$ by any of these factors results in a number outside of $\mathbb{\mathbb{Z}}[\sqrt{30}]$. Likewise, dividing any of these factors (other than the unit) by $\sqrt{30}$ also results in a number outside of $\mathbb{\mathbb{Z}}[\sqrt{30}]$.

Ideals really help us make sense of multiple distinct factorizations in these domains.

$p$	Factorization of $⟨p⟩$
	In $\mathbb{\mathbb{Z}}[\sqrt{-30}]$	In $\mathbb{\mathbb{Z}}[\sqrt{30}]$
2	$⟨2,\sqrt{-30}{⟩}^2$	$⟨2,\sqrt{30}{⟩}^2$
3	$⟨3,\sqrt{-30}{⟩}^2$	$⟨3,\sqrt{30}{⟩}^2$
5	$⟨5,\sqrt{-30}{⟩}^2$	$⟨5,\sqrt{30}{⟩}^2$
7	Prime	$⟨7,3-\sqrt{30}⟩⟨7,3+\sqrt{30}⟩$
11	$⟨11,5-\sqrt{-30}⟩⟨11,5+\sqrt{-30}⟩$	Prime
13	$⟨13,3-\sqrt{-30}⟩⟨13,3+\sqrt{-30}⟩$	$⟨13,2-\sqrt{30}⟩⟨13,2+\sqrt{30}⟩$
17	$⟨17,2-\sqrt{-30}⟩⟨17,2+\sqrt{-30}⟩$	$⟨17,8-\sqrt{30}⟩⟨17,8+\sqrt{30}⟩$
19	Prime	$⟨7-\sqrt{30}⟩⟨7+\sqrt{30}⟩$
23	$⟨23,4-\sqrt{-30}⟩⟨23,4+\sqrt{-30}⟩$	Prime
29	$⟨29,12-\sqrt{-30}⟩⟨29,12+\sqrt{-30}⟩$	$⟨1-\sqrt{30}⟩⟨1+\sqrt{30}⟩$
31	$⟨1-\sqrt{-30}⟩⟨1+\sqrt{-30}⟩$	Prime
37	$⟨37,9-\sqrt{-30}⟩⟨37,9+\sqrt{-30}⟩$
41
43
47

Factorization of 30 in some quadratic integer rings

In $\mathbb{\mathbb{Z}}$, 30 is the product of three primes. But it has different factorizations in many quadratic integer rings.

$\mathbb{\mathbb{Z}}[i]$	$(1\pm i)3(2\pm i)$
$\mathbb{\mathbb{Z}}[\sqrt{-2}]$	$(-1)(\sqrt{-2}{)}^2(1\pm \sqrt{-2})5$	$\mathbb{\mathbb{Z}}[\sqrt{2}]$	$(\sqrt{2}{)}^{2}3\times 5$
$\mathbb{\mathbb{Z}}[\omega ]$	$(-1)2(1+2\omega {)}^{2}5$	$\mathbb{\mathbb{Z}}[\sqrt{3}]$	$(-1)(1\pm \sqrt{3})(\sqrt{3}{)}^{2}5$
$\mathbb{\mathbb{Z}}[\sqrt{-5}]$	$(-1)2\times 3(\sqrt{-5}{)}^2$	$\mathbb{\mathbb{Z}}[\varphi ]$	$2\times 3(-1+2\varphi {)}^2$
$\mathbb{\mathbb{Z}}[\sqrt{-6}]$	2 × 3 × 5	$\mathbb{\mathbb{Z}}[\sqrt{6}]$	$(2\pm \sqrt{6})(3\pm \sqrt{6})(1\pm \sqrt{6})$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$	$(\frac{1}{2}\pm \frac{\sqrt{-7}}{2})3\times 5$	$\mathbb{\mathbb{Z}}[\sqrt{7}]$	$(-1)(3\pm \sqrt{7})(2\pm \sqrt{7})5$
$\mathbb{\mathbb{Z}}[\sqrt{-10}]$	2 × 3 × 5	$\mathbb{\mathbb{Z}}[\sqrt{10}]$	2 × 3 × 5
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-11})}$	$2(\frac{1}{2}\pm \frac{\sqrt{-11}}{2})(\frac{3}{2}\pm \frac{\sqrt{-11}}{2})$	$\mathbb{\mathbb{Z}}[\sqrt{11}]$	$(-1)(3\pm \sqrt{11})3(4\pm \sqrt{11})$
$\mathbb{\mathbb{Z}}[\sqrt{-13}]$	2 × 3 × 5	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{13})}$	$(-1)2(\frac{1}{2}\pm \frac{\sqrt{13}}{2})5$
$\mathbb{\mathbb{Z}}[\sqrt{-14}]$	2 × 3 × 5 OR $(4-\sqrt{-14})(4+\sqrt{-14})$	$\mathbb{\mathbb{Z}}[\sqrt{14}]$	$(-1)(4\pm \sqrt{14})3(3\pm \sqrt{14})$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-15})}$	2 × 3 × 5	$\mathbb{\mathbb{Z}}[\sqrt{15}]$	2 × 3 × 5
$\mathbb{\mathbb{Z}}[\sqrt{-17}]$		${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$	$(-1)(\frac{3}{2}\pm \frac{\sqrt{17}}{2})3\times 5$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-19})}$	$2\times 3(\frac{1}{2}-\frac{\sqrt{-19}}{2})(\frac{1}{2}+\frac{\sqrt{-19}}{2})$	$\mathbb{\mathbb{Z}}[\sqrt{19}]$	$(13\pm 3\sqrt{19})(4\pm \sqrt{19})(9\pm 2\sqrt{19})$

Representation of 30 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	11110	1010	132	110	50	42	36	33	30	28	26	24	22	20	1E	1D	1C	1B	1A

This number is palindromic in bases 9, 14 and 29, and trivially so in base 31 and higher.

It is a Harshad number nontrivially in bases 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 16, 21, 25, 26, 28, 29.

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