105

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105 is an integer, the largest known $n$ such that $n-2^{k}$ is prime for all $1<k<\log _{2}n$ (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

Values for number theoretic functions with 105 as an argument

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

Factorization of 105 in some quadratic integer rings

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

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