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# 36

Please do not rely on any information it contains.

36 is an integer. As the square of 6, it is the smallest nontrivial square to also be a triangular number.

## 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.

 ${\sqrt {36}}$ 6.00000000 36 2 1296 ${\sqrt[{3}]{36}}$ 3.30192724 A010607 36 3 46656 ${\sqrt[{4}]{36}}$ 2.44948974 A010464 36 4 1679616 ${\sqrt[{5}]{36}}$ 2.04767251 A011121 36 5 60466176 ${\sqrt[{6}]{36}}$ 1.81712059 A005486 36 6 16777216 ${\sqrt[{7}]{36}}$ 1.66851044 36 7 78364164096 ${\sqrt[{8}]{36}}$ 1.56508458 A011004 36 8 2821109907456 ${\sqrt[{9}]{36}}$ 1.48909532 36 9 101559956668416 ${\sqrt[{10}]{36}}$ 1.43096908 A011091 36 10 3656158440062976 ${\sqrt[{11}]{36}}$ 1.38510292 36 11 131621703842267136 ${\sqrt[{12}]{36}}$ 1.34800615 A011215 36 12 4738381338321616896 A009980

## Logarithms and 36th powers

In the OEIS specifically and mathematics in general, $\log x$ refers to the natural logarithm of $x$ , whereas all other bases are specified with a subscript.

From Fermat's little theorem it follows that if $b$ is coprime to 37, then $b^{36}\equiv 1\mod 37$ .

If $n$ is not a multiple of 73, then either $n^{36}-1$ or $n^{36}+1$ is. Hence the formula for the Legendre symbol $\left({\frac {a}{73}}\right)=a^{36}\mod 73$ .

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

 $\mu (36)$ 0 $M(36)$ −1 $\pi (36)$ 11 $\sigma _{1}(36)$ $\sigma _{0}(36)$ $\phi (36)$ 12 $\Omega (36)$ 4 $\omega (36)$ 2 $\lambda (36)$ This is the Carmichael lambda function. $\lambda (36)$ This is the Liouville lambda function. 36! 371993326789901217467999448150835200000000 $\Gamma (36)$ 10333147966386144929666651337523200000000

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.

 $-1$ 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