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# 105

Please do not rely on any information it contains.

105 is an integer, the largest known $n$ such that $n-2^{k}$ is prime for all $1 (see A039669).

## Membership in core sequences

 Odd numbers ..., 99, 101, 103, 105, 107, 109, 111, ... A005408 Composite numbers ..., 100, 102, 104, 105, 106, 108, 110, ... A002808 Squarefree numbers ..., 101, 102, 103, 105, 106, 107, 109, ... A005117 Triangular numbers ..., 66, 78, 91, 105, 120, 136, 153, ... A000217 Double factorials ..., 8, 15, 48, 105, 384, 945, 3840, ... A006882 Lucky numbers ..., 87, 93, 99, 105, 111, 115, 127, ... A000959

## Sequences pertaining to 105

 Divisors of 105 1, 3, 5, 7, 15, 21, 35, 105 A018286 Multiples of 105 0, 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050, 1155, ...

## Partitions of 105

There are 342325709 partitions of 105.

## Roots and powers of 105

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

TABLE GOES HERE

PLACEHOLDER

## Factorization of some small integers in a quadratic integer ring adjoining −105, 105

Neither $\mathbb {Z} [{\sqrt {-105}}]$ nor ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {105}})}$ are unique factorization domains. Units in ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {105}})}$ are of the form $(41+4{\sqrt {105}})^{n}$ .

Since 105 = 3 × 5 × 7, it follows that those primes having a least significant digit of 3 or 7 in base 10 are inert and irreducible in these domains. But ending in 1 or 9 does not automatically guarantee the prime splits in either of these domains.

 $n$ $\mathbb {Z} [{\sqrt {-105}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {105}})}$ 2 Irreducible 3 Irreducible despite indication of ramification 4 2 2 2 2 OR $\left({\frac {11}{2}}-{\frac {\sqrt {105}}{2}}\right)\left({\frac {11}{2}}+{\frac {\sqrt {105}}{2}}\right)$ 5 Irreducible despite indication of ramification $(-1)(10-{\sqrt {105}})(10+{\sqrt {105}})$ 6 2 × 3 2 × 3 OR $(-1)\left({\frac {9}{2}}-{\frac {\sqrt {105}}{2}}\right)\left({\frac {9}{2}}+{\frac {\sqrt {105}}{2}}\right)$ 7 Irreducible despite indication of ramification 8 2 3 9 3 2 10 2 × 5 $(-1)2(10-{\sqrt {105}})(10+{\sqrt {105}})$ 11 Irreducible Prime 12 2 2 × 3 13 Irreducible Irreducible despite positive Legendre symbol 14 2 × 7 15 3 × 5 $(-1)3(10-{\sqrt {105}})(10+{\sqrt {105}})$ 16 2 4 17 Prime 18 2 × 3 2 19 Irreducible Prime 20 2 2 × 5 $(-1)2^{2}(10-{\sqrt {105}})(10+{\sqrt {105}})$ Ideals help us make sense of these distinct factorizatons.

TABLE GOES HERE

PLACEHOLDER

TABLE GOES HERE

## Representation of 105 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1101001 10220 1221 410 253 210 151 126 105 96 89 81 77 70 69 63 5F 5A 55

 $-1$ 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