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# 33

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, ... ${\displaystyle 3x+1}$ 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, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.

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

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

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PLACEHOLDER

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## 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.

 ${\displaystyle -1}$ 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