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# 85

Please do not rely on any information it contains.

85 is an integer. The 85th square pyramidal number, 208335, is also a triangular number, the largest to have this property (see A053611).

## Membership in core sequences

 Odd numbers ..., 79, 81, 83, 85, 87, 89, 91, ... A005408 Squarefree numbers ..., 79, 82, 83, 85, 86, 87, 89, ... A005117 Semiprimes ..., 74, 77, 82, 85, 86, 87, 91, ... A001358 Composite numbers ..., 81, 82, 84, 85, 86, 87, 88, ... A002808 Numbers that are the sum of 2 squares ..., 80, 81, 82, 85, 89, 90, 97, ... A001481 Jacobsthal numbers ..., 11, 21, 43, 85, 171, 341, ... A001045

## Sequences pertaining to 85

 Multiples of 85 0, 85, 170, 255, 340, 425, 510, 595, 680, 765, 850, 935, ... ${\displaystyle 3x+1}$ sequence starting at 75 75, 226, 113, 340, 170, 85, 256, 128, 64, 32, 16, 8, 4, 2, ... A258056

## Partitions of 85

There are 30167357 partitions of 85.

There is only one way to represent 85 as a sum of two primes, 2 + 83, but there are many ways to represent 85 as a sum of three distinct primes: 73 + 7 + 5 = 71 + 11 + 3 = 67 + 13 + 5 = 67 + 11 + 7 = 61 + 19 + 5 = 61 + 17 + 7 = 61 + 13 + 11 = 59 + 23 + 3 = 59 + 19 + 7 = 53 + 29 + 3 = 53 + 19 + 13 = 47 + 31 + 7 = 43 + 37 + 5 = 43 + 31 + 11 = 43 + 29 + 13 = 43 + 23 + 19 = 41 + 37 + 7 = 41 + 31 + 13 = 37 + 31 + 17 = 37 + 29 + 19 = 85.

PLACEHOLDER

PLACEHOLDER

## Values for number theoretic functions with 85 as an argument

 ${\displaystyle \mu (85)}$ 1 ${\displaystyle M(85)}$ ${\displaystyle \pi (85)}$ ${\displaystyle \sigma _{1}(85)}$ ${\displaystyle \sigma _{0}(85)}$ 4 ${\displaystyle \phi (85)}$ 64 ${\displaystyle \Omega (85)}$ 2 ${\displaystyle \omega (85)}$ ${\displaystyle \lambda (85)}$ This is the Carmichael lambda function. ${\displaystyle \lambda (85)}$ This is the Liouville lambda function. 85! ${\displaystyle \Gamma (85)}$

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −85, 85

Neither ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {85}})}}$ nor ${\displaystyle \mathbb {Z} [{\sqrt {-85}}]}$ are unique factorization domains. Units in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {85}})}}$ are of the form ${\displaystyle \left({\frac {9}{2}}+{\frac {\sqrt {85}}{2}}\right)^{n}}$.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-85}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {85}})}}$ 2 Irreducible Prime 3 Prime Irreducible despite positive Legendre symbol 4 2 2 5 Irreducible despite indication of ramification 6 2 × 3 7 Prime Irreducible despite positive Legendre symbol 8 2 3 9 3 2 3 2 OR ${\displaystyle (-1)\left({\frac {7}{2}}-{\frac {\sqrt {85}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {85}}{2}}\right)}$ 10 2 × 5 11 Irreducible 12 2 2 × 3 13 Prime 14 2 × 7 15 3 × 5 3 × 5 OR ${\displaystyle (-1)\left({\frac {5}{2}}-{\frac {\sqrt {85}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {85}}{2}}\right)}$ 16 2 4 17 Irreducible despite indication of ramification 18 2 × 3 2 19 ${\displaystyle (-1)\left({\frac {3}{2}}-{\frac {\sqrt {85}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {85}}{2}}\right)}$ 20 2 2 × 5 21 3 × 7 3 × 7 OR ${\displaystyle (-1)\left({\frac {1}{2}}-{\frac {\sqrt {85}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {85}}{2}}\right)}$

To drive home the point that ${\displaystyle \mathbb {Z} [{\sqrt {-85}}]}$ has class number 4, we'll show a few more numbers which not only have more than one distinct factorization, but the distinct factorizations have a different number of irreducible factors.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-85}}]}$ 110 2 × 5 × 11 OR ${\displaystyle (5-{\sqrt {-85}})(5+{\sqrt {-85}})}$ 310 2 × 5 × 31 OR ${\displaystyle (15-{\sqrt {-85}})(15+{\sqrt {-85}})}$ 374 2 × 11 × 17 OR ${\displaystyle (17-{\sqrt {-85}})(17+{\sqrt {-85}})}$ 710 2 × 5 × 71 OR ${\displaystyle (25-{\sqrt {-85}})(25+{\sqrt {-85}})}$

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

 ${\displaystyle p}$ Factorization of ${\displaystyle \langle p\rangle }$ In ${\displaystyle \mathbb {Z} [{\sqrt {-85}}]}$ In ${\displaystyle \mathbb {Z} [{\sqrt {85}}]}$ 2 ${\displaystyle \langle 2,1+{\sqrt {-85}}\rangle ^{2}}$ Prime 3 Prime ${\displaystyle \langle 3,1-{\sqrt {85}}\rangle \langle 3,1+{\sqrt {85}}\rangle }$ 5 ${\displaystyle \langle 5,{\sqrt {-85}}\rangle ^{2}}$ ${\displaystyle \langle 5,{\sqrt {85}}\rangle ^{2}}$ 7 Prime ${\displaystyle \langle 7,1-{\sqrt {85}}\rangle \langle 7,1+{\sqrt {85}}\rangle }$ 11 ${\displaystyle \langle 11,5-{\sqrt {-85}}\rangle \langle 11,5+{\sqrt {-85}}\rangle }$ Prime 13 Prime 17 ${\displaystyle \langle 17,{\sqrt {-85}}\rangle ^{2}}$ ${\displaystyle \langle 17,{\sqrt {85}}\rangle ^{2}}$ 19 23 29 31 37 41 43 47

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## Representation of 85 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1010101 10011 1111 320 221 151 125 104 85 78 71 67 61 5A 55 50 4D 49 45

Note that 85 is palindromic in binary (see A006995). It is also palindromic in bases 4, 21, 84 and trivially so in bases 86 and higher.

 ${\displaystyle -1}$ 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