33

This article is under construction.            

Please do not rely on any information it contains.            

33 is an integer, the largest that is not a sum of distinct triangular numbers.

Membership in core sequences

Odd numbers	..., 27, 29, 31, 33, 35, 37, 39, ...	A005408
Semiprimes	..., 22, 25, 26, 33, 34, 35, 38, ...	A001358
Squarefree numbers	..., 29, 30, 31, 33, 34, 35, 37, ...	A005117
Composite numbers	..., 28, 30, 32, 33, 34, 35, 36, ...	A002808
Deficient numbers	..., 29, 31, 32, 33, 34, 35, 37, ...	A005100
Lucky numbers	..., 21, 25, 31, 33, 37, 43, 49, ...	A000959

Sequences pertaining to 33

Multiples of 33	0, 33, 66, 99, 132, 165, 198, 231, 264, 297, 330, 363, 396, ...
$3x+1$ sequence starting at 33	33, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, ...	A008880

Partitions of 33

There are 10143 partitions of 33.

Although there are no partitions of 33 into distinct triangular numbers, there are plenty of triangular number partitions with repeated numbers, such as: 28 + 3 + 1 + 1, 21 + 10 + 1 + 1, 15 + 15 + 3, etc.

Roots and powers of 33

In the table below, irrational numbers are given truncated to eight decimal places.

TABLE GOES HERE

Logarithms and 33rd 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.

If $n$ is not a multiple of 67, then either $n^{33}-1$ or $n^{33}+1$ is. Hence the formula for the Legendre symbol $\left({\frac {a}{67}}\right)=a^{33}\mod 67$.

As above, irrational numbers in the following table are truncated to eight decimal places.

TABLE GOES HERE

Values for number theoretic functions with 33 as an argument

TABLE GOES HERE

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −33, 33

TABLE GOES HERE

Factorization of 33 in some quadratic integer rings

PLACEHOLDER

TABLE GOES HERE

Representation of 33 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	100001	1020	201	113	53	45	41	36	33	30	29	27	25	23	21	1G	1F	1C	1D

33 is palindromic in binary, decimal and base 32, and trivially palindromic in base 34 and higher.

See also

Some integers

							$-1$
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