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# 53

Please do not rely on any information it contains.

53 is an integer.

## Membership in core sequences

 Odd numbers ..., 47, 49, 51, 53, 55, 57, 59, ... A005408 Prime numbers ..., 41, 43, 47, 53, 59, 61, ... A000040 Squarefree numbers ..., 46, 47, 51, 53, 55, 57, 58, ... A005117

In Pascal's triangle, 53 occurs twice. (In Lozanić's triangle, 53 occurs four times).

## Sequences pertaining to 53

 Multiples of 53 0, 53, 106, 159, 212, 265, 318, 371, 424, 477, 530, 583, 636, ... ${\displaystyle 3x+1}$ sequence beginning at 15 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, ... A033480

## Partitions of 53

There are 329931 partitions of 53. Of these, the ones consisting of distinct which the of the [FINISH WRITING]

## Roots and powers of 53

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

TABLE GOES HERE

## Logarithms and 53rd 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.

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

TABLE GOES HERE

See A089081 for the 26th powers of integers.

TABLE GOES HERE

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −53, 53

The commutative quadratic integer ring with unity ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {53}}]\,}$, with units of the form ${\displaystyle \scriptstyle \pm \left({\frac {7}{2}}+{\frac {\sqrt {53}}{2}}\right)^{n}\,}$ (${\displaystyle \scriptstyle n\,\in \,\mathbb {Z} \,}$), is a unique factorization domain.

TABLE GOES HERE

${\displaystyle \mathbb {Z} [{\sqrt {-53}}]}$ is not a unique factorization domain either. But the range of 1 through 20 does not provide as interesting a window for the of the [FINISH WRITING]

## Factorization of 53 in some quadratic integer rings

As was mentioned above, 26 is the product of two primes in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings.

TABLE

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## Representation of 53 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 110101 1222 311 203 125 104 65 58 53 49 45 41 3B 38 35 32 2H 2F 2D

Note that the hexadecimal digits of 53 are the same as its decimal digits, just reversed.

Far more interesting than that, however, is the fact that 53 is not palindromic in any of the bases shown here. In fact, it's not palindromic in any base from binary to base 51. This means that 53 is a strictly non-palindromic number (see A016038).

 ${\displaystyle -1}$ 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