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85
85 is an integer. The 85th square pyramidal number, 208335, is also a triangular number, the largest to have this property (see A053611).
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 85
- 3 Partitions of 85
- 4 Roots and powers of 85
- 5 Logarithms and 85th powers
- 6 Values for number theoretic functions with 85 as an argument
- 7 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −85, 85
- 8 Factorization of 85 in some quadratic integer rings
- 9 Representation of 85 in various bases
- 10 See also
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
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 |