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# 125

Please do not rely on any information it contains.

125 is an integer. It is the only number with a base 10 representation that contains the base 10 representations of its divisors (1, 5, 25) as proper substrings.

## Membership in core sequences

 Odd numbers ..., 119, 125, 123, 125, 127, 129, 131, ... A005408 Composite numbers ..., 122, 123, 124, 125, 126, 128, 129, ... A002808 Perfect cubes ..., 8, 27, 64, 125, 216, 343, 512, ... A000578 Number of trees on $n$ labeled nodes 1, 1, 1, 3, 16, 125, 1296, 16807, 262144, ... A000124 Numbers that are the sum of 2 squares ..., 117, 121, 122, 125, 128, 130, 136, ... A001481

## Sequences pertaining to 125

 Multiples of 125 125, 250, 375, 500, 625, 750, 875, 1000, 1125, 1250, ... $3x+1$ sequence starting at 97 ..., 83, 250, 125, 376, 188, 94, 47, 142, 71, 214, ... A008873

## Partitions of 125

There are 3163127352 partitions of 125.

Since 123 is composite, any prime partition of 125 must consist of at least three parts.

## Roots and powers of 125

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

TABLE GOES HERE

## Logarithms and 125th 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.

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

TABLE GOES HERE

TABLE GOES HERE

## Factorization of 125 in some quadratic integer rings

As was mentioned above, 125 is the cube of 5. But it has different factorizations in some quadratic integer rings, and in a few cases it's not a simple matter of taking the factorization of 5 and adding in some exponent 3s.

 $\mathbb {Z} [i]$ $(2-i)^{3}(2+i)^{3}$ $\mathbb {Z} [{\sqrt {-2}}]$ 5 3 $\mathbb {Z} [{\sqrt {2}}]$ 5 3 $\mathbb {Z} [\omega ]$ $\mathbb {Z} [{\sqrt {3}}]$ $\mathbb {Z} [{\sqrt {-5}}]$ $(-1)({\sqrt {-5}})^{6}$ $\mathbb {Z} [\phi ]$ $(-1+2\phi )^{6}$ $\mathbb {Z} [{\sqrt {-6}}]$ 5 3 $\mathbb {Z} [{\sqrt {6}}]$ $(-1)(1-{\sqrt {6}})^{3}(1+{\sqrt {6}})^{3}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$ $\mathbb {Z} [{\sqrt {7}}]$ 5 3 $\mathbb {Z} [{\sqrt {-10}}]$ $\mathbb {Z} [{\sqrt {10}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$ $\left({\frac {3}{2}}-{\frac {\sqrt {-11}}{2}}\right)^{3}\left({\frac {3}{2}}+{\frac {\sqrt {-11}}{2}}\right)^{3}$ $\mathbb {Z} [{\sqrt {11}}]$ $(4-{\sqrt {11}})^{3}(4+{\sqrt {11}})^{3}$ $\mathbb {Z} [{\sqrt {-13}}]$ 5 3 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$ 5 3 $\mathbb {Z} [{\sqrt {-14}}]$ $\mathbb {Z} [{\sqrt {14}}]$ $(-1)(3-{\sqrt {14}})^{3}(3+{\sqrt {14}})^{3}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ $\mathbb {Z} [{\sqrt {15}}]$ 5 3 $\mathbb {Z} [{\sqrt {-17}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$ $\left({\frac {1}{2}}-{\frac {\sqrt {-19}}{2}}\right)^{3}\left({\frac {1}{2}}+{\frac {\sqrt {-19}}{2}}\right)^{3}$ $\mathbb {Z} [{\sqrt {19}}]$ $(9-2{\sqrt {19}})^{3}(9+2{\sqrt {19}})^{3}$ ## Representation of 125 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1111101 11122 1331 1000 325 236 175 148 125 104 A5 98 8D 85 7D 76 6G 6B 65

In decimal, 125 is a Friedman number, since $5^{2+1}=125$ .

 $-1$ 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