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# 30

Please do not rely on any information it contains.

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 ${\displaystyle 3x+1}$ 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.

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

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

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

 ${\displaystyle \mu (30)}$ −1 ${\displaystyle M(30)}$ −3 ${\displaystyle \pi (30)}$ 8 ${\displaystyle \sigma _{1}(30)}$ 72 ${\displaystyle \sigma _{0}(30)}$ 8 ${\displaystyle \phi (30)}$ 8 ${\displaystyle \Omega (30)}$ 3 ${\displaystyle \omega (30)}$ 3 ${\displaystyle \lambda (30)}$ 4 This is the Carmichael lambda function. ${\displaystyle \lambda (30)}$ −1 This is the Liouville lambda function. 30! 265252859812191058636308480000000 ${\displaystyle \Gamma (30)}$ 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 ${\displaystyle \mathbb {Z} [{\sqrt {30}}]}$, with units of the form ${\displaystyle \pm (11+2{\sqrt {30}})^{n}\,}$ (${\displaystyle n\in \mathbb {Z} }$), 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 ${\displaystyle \mathbb {Z} [{\sqrt {30}}]}$. But ending in 1 or 9 does not automatically guarantee the prime splits in ${\displaystyle \mathbb {Z} [{\sqrt {30}}]}$.

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

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

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

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

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

## Factorization of 30 in some quadratic integer rings

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

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

## 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.

 ${\displaystyle -1}$ 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