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85

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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, ...
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.

Roots and powers of 85

PLACEHOLDER

Logarithms and 85th powers

PLACEHOLDER

Values for number theoretic functions with 85 as an argument

1
4
64
2
This is the Carmichael lambda function.
This is the Liouville lambda function.
85!

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

Neither nor are unique factorization domains. Units in are of the form .

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
10 2 × 5
11 Irreducible
12 2 2 × 3
13 Prime
14 2 × 7
15 3 × 5 3 × 5 OR
16 2 4
17 Irreducible despite indication of ramification
18 2 × 3 2
19
20 2 2 × 5
21 3 × 7 3 × 7 OR

To drive home the point that 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.

110 2 × 5 × 11 OR
310 2 × 5 × 31 OR
374 2 × 11 × 17 OR
710 2 × 5 × 71 OR

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

Factorization of
In In
2 Prime
3 Prime
5
7 Prime
11 Prime
13 Prime
17
19
23
29
31
37
41
43
47

Factorization of 85 in some quadratic integer rings

PLACEHOLDER

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.

See also

Some integers
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