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# 70

Please do not rely on any information it contains.

70 is an integer.

## 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

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 $\langle p\rangle$ 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

PLACEHOLDER

TABLE GOES HERE

## Representation of 70 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1000110 2221 1012 240 154 130 106 77 70 64 5A 55 4E 4A 46 42 3G 3D 3A

REMARKS GO HERE

 $-1$ 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