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36
36 is an integer. As the square of 6, it is the smallest nontrivial square to also be a triangular number.
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 36
- 3 Partitions of 36
- 4 Roots and powers of 36
- 5 Logarithms and 36th powers
- 6 Values for number theoretic functions with 36 as an argument
- 7 Factorization of 36 in some quadratic integer rings
- 8 Representation of 36 in various bases
- 9 See also
- 10 References
Membership in core sequences
Even numbers | ..., 30, 32, 34, 36, 38, 40, 42, ... | A005843 |
Composite numbers | ..., 33, 34, 35, 36, 38, 39, 40, ... | A002808 |
Abundant numbers | 12, 18, 20, 24, 30, 36, 40, 42, ... | A005101 |
Triangular numbers | ..., 15, 21, 28, 36, 45, 55, 66, ... | A000217 |
Perfect squares | 1, 4, 9, 16, 25, 36, 49, 64, 81, ... | A000290 |
Loeschian numbers | ..., 27, 28, 31, 36, 37, 39, 43, ... | A003136 |
Sequences pertaining to 36
Multiples of 36 | 0, 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, ... | A044102 |
Divisors of 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | A018256 |
Partitions of 36
There are 17977 partitions of 36.
The Goldbach representations of 36 are 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19.
Roots and powers of 36
In the table below, irrational numbers are given truncated to eight decimal places.
6.00000000 | 36 2 | 1296 | ||
3.30192724 | A010607 | 36 3 | 46656 | |
2.44948974 | A010464 | 36 4 | 1679616 | |
2.04767251 | A011121 | 36 5 | 60466176 | |
1.81712059 | A005486 | 36 6 | 16777216 | |
1.66851044 | 36 7 | 78364164096 | ||
1.56508458 | A011004 | 36 8 | 2821109907456 | |
1.48909532 | 36 9 | 101559956668416 | ||
1.43096908 | A011091 | 36 10 | 3656158440062976 | |
1.38510292 | 36 11 | 131621703842267136 | ||
1.34800615 | A011215 | 36 12 | 4738381338321616896 | |
A009980 |
Logarithms and 36th powers
In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.
From Fermat's little theorem it follows that if is coprime to 37, then .
If is not a multiple of 73, then either or is. Hence the formula for the Legendre symbol .
As above, irrational numbers in the following table are truncated to eight decimal places.
TABLE GOES HERE
Values for number theoretic functions with 36 as an argument
0 | ||
−1 | ||
11 | ||
12 | ||
4 | ||
2 | ||
This is the Carmichael lambda function. | ||
This is the Liouville lambda function. | ||
36! | 371993326789901217467999448150835200000000 | |
10333147966386144929666651337523200000000 |
Factorization of 36 in some quadratic integer rings
TABLE GOES HERE
Representation of 36 in various bases
Base | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
Representation | 100100 | 1100 | 210 | 121 | 100 | 51 | 44 | 40 | 36 | 33 | 30 | 2A | 28 | 26 | 24 | 22 | 20 | 1H | 1G |
In theory, any integer greater than 1 can be used as a base for a positional numeral system. In practice, however, the large number of symbols required keeps bases beyond sexagesimal firmly in the realm of theory.
If we use letters of the Latin alphabet to represent digits above 9, then we have A = 9 + 1, B = 9 + 2, C = 9 + 3, and so on and so forth to Z = 9 + 26. For this scheme, then base 36 is the largest base that can be used. It is possible to extend this to go further, of course, such as by distinguishing between uppercase and lowercase letters (e.g., A could correspond to 10, a could correspond to 36).
36 is a Harshad number in bases 2 through 13, except for base 7.
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 |