70

This article is under construction.

Please do not rely on any information it contains.

70 is an integer.

1 Membership in core sequences
2 Sequences pertaining to 70
3 Partitions of 70
4 Roots and powers of 70
5 Logarithms and seventieth powers
6 Values for number theoretic functions with 70 as an argument
7 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −70, 70
8 Factorization of 70 in some quadratic integer rings
9 Representation of 70 in various bases
10 See also

Membership in core sequences

Even numbers	..., 64, 66, 68, 70, 72, 74, 76, ...	A005843
Composite numbers	..., 66, 68, 69, 70, 72, 74, 75, ...	A002808
Squarefree numbers	..., 66, 67, 69, 70, 71, 73, 74, ...	A005117
Abundant numbers	..., 56, 60, 66, 70, 72, 78, 80, ...	A005101
Central binomial coefficients	1, 2, 6, 20, 70, 252, 924, 3432, ...	A000984
Pell numbers	1, 2, 5, 12, 29, 70, 169, 408, ...	A000129
Pentagonal numbers	1, 5, 12, 22, 35, 51, 70, 92, 117, ...	A000326

In Pascal's triangle, 70 occurs thrice, the first time as a central binomial coefficient (A000984) in the eighth row.

Sequences pertaining to 70

Multiples of 70	70, 140, 210, 280, 350, 420, 490, 560, 630, 700, 770, 840, ...
Divisors of 70	1, 2, 5, 7, 10, 14, 35, 70	A018270
$3x+1$ sequence beginning at 15	15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ...	A033480
$3x-1$ sequence beginning at 63	63, 188, 94, 47, 140, 70, 35, 104, 52, 26, 13, 38, 19, 56, ...	A008895

Partitions of 70

There are 4087968 partitions of 70.

The Goldbach representations of 70 are: 67 + 3 = 59 + 11 = 53 + 17 = 47 + 23 = 41 + 29 = 70.

Since the proper divisors of 70 add up to 74, 70 is an abundant number. The reason we bring this up in connection to partitions is that most abundant numbers have at least one partition consisting of distinct proper divisors. But 70 has no such partition, and hence it is a weird number. It does have partitions consisting solely of proper divisors, but in each case at least one proper divisor is repeated, e.g., 1 + 1 + 2 + 7 + 10 + 14 + 35 = 70.

Roots and powers of 70

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

TABLE GOES HERE

Logarithms and seventieth powers

REMARKS

TABLE

Values for number theoretic functions with 70 as an argument

$\mu (70)$	–1
$M(70)$	–2
$\pi (70)$	19
$\sigma _{1}(70)$	74
$\sigma _{0}(70)$	8
$\phi (70)$	24
$\Omega (70)$	3
$\omega (70)$	3
$\lambda (70)$	12	This is the Carmichael lambda function.
$\lambda (70)$	–1	This is the Liouville lambda function.
70!	1.19785716... × 10 100
$\Gamma (70)$	1.71122452... × 10 98

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −70, 70

The commutative quadratic integer ring with unity $\mathbb {Z} [{\sqrt {70}}]$ , with units of the form $\pm (251+30{\sqrt {70}})^{n}\,$ ( $n\in \mathbb {Z}$ ), is not a unique factorization domain. But since 70 = 7 × 10, it follows that those primes having a least significant digit of 3 or 7 in base 10 are inert and irreducible in $\mathbb {Z} [{\sqrt {70}}]$ . But ending in 1 or 9 does not automatically guarantee the prime splits in $\mathbb {Z} [{\sqrt {70}}]$ .

$\mathbb {Z} [{\sqrt {-70}}]$ is not a unique factorization domain either. However, its scarcity of units gives us greater confidence in identifying instances of non-unique factorization.

$n$	$\mathbb {Z} [{\sqrt {-70}}]$	$\mathbb {Z} [{\sqrt {70}}]$
2	Irreducible
3	Prime	Irreducible
4	2 2
5	Irreducible	$(-1)(25-3{\sqrt {70}})(25+3{\sqrt {70}})$
6	2 × 3
7	Irreducible
8	2 3
9	3 2	3 2 OR $(17-2{\sqrt {70}})(17+2{\sqrt {70}})$
10	2 × 5	$(-1)2(25-3{\sqrt {70}})(25+3{\sqrt {70}})$
11	Irreducible	$(9-{\sqrt {70}})(9+{\sqrt {70}})$
12	2 2 × 3
13	Prime
14	2 × 7	2 × 7 OR $(42-5{\sqrt {70}})(42+5{\sqrt {70}})$
15	3 × 5	$(-1)3(25-3{\sqrt {70}})(25+3{\sqrt {70}})$
16	2 4
17	Irreducible
18	2 × 3 2
19	Irreducible	Prime
20	2 2 × 5	$(-1)2^{2}(25-3{\sqrt {70}})(25+3{\sqrt {70}})$

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

$p$	Factorization of $\langle p\rangle$
$p$	In $\mathbb {Z} [{\sqrt {-70}}]$	In $\mathbb {Z} [{\sqrt {70}}]$
2	$\langle 2,{\sqrt {-70}}\rangle ^{2}$	$\langle 2,{\sqrt {70}}\rangle ^{2}$
3	Prime	$\langle 3,1-{\sqrt {70}}\rangle \langle 3,1+{\sqrt {70}}\rangle$
5	$\langle 5,{\sqrt {-70}}\rangle ^{2}$	$\langle 5,{\sqrt {70}}\rangle ^{2}$
7	$\langle 7,{\sqrt {-70}}\rangle ^{2}$	$\langle 7,{\sqrt {70}}\rangle ^{2}$
11	Prime	$\langle 9-{\sqrt {70}}\rangle \langle 9+{\sqrt {70}}\rangle$
13	Prime
17	$\langle 17,7-{\sqrt {-70}}\rangle \langle 17,7+{\sqrt {-70}}\rangle$	$\langle 17,6-{\sqrt {70}}\rangle \langle 17,6+{\sqrt {70}}\rangle$
19	$\langle 19,5-{\sqrt {-70}}\rangle \langle 19,5+{\sqrt {-70}}\rangle$	Prime
23
29
31
37
41
43
47