15

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15 is a composite number. In fact, it is the smallest composite number $n$ with the property that there is only one group of order $n$ (see A000001 and A050384).

Membership in core sequences

Odd numbers	..., 9, 11, 13, 15, 17, 19, 21, ...	A005408
Composite numbers	..., 10, 12, 14, 15, 16, 18, ...	A002808
Semiprimes	4, 6, 9, 10, 14, 15, 21, 22, ...	A001358
Partition numbers	1, 1, 2, 3, 5, 7, 11, 15, 22, ...	A000041
Triangular numbers	1, 3, 6, 10, 15, 21, 28, 36, ...	A000217

In Pascal's triangle, 15 occurs four times, the first two times surrounding the first occurrence of 20, with 6 on either side.

Sequences pertaining to 15

Multiples of 15	15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, ...	A008597
15-gonal numbers	1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, ...	A051867
$3x+1$ sequence starting at 15	15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ...	A033480

Partitions of 15

There are 176 partitions of 15.

The Goldbach representations of 15 using distinct primes are: 2 + 13 = 3 + 5 + 7.

Roots and powers of 15

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

${\sqrt {15}}$	3.87298334	A010472	15 2	225
${\sqrt[{3}]{15}}$	2.46621207	A010587	15 3	3375
${\sqrt[{4}]{15}}$	1.96798967	A011012	15 4	50625
${\sqrt[{5}]{15}}$	1.71877192	A011100	15 5	759375
${\sqrt[{6}]{15}}$	1.57041780	A011350	15 6	11390625
${\sqrt[{7}]{15}}$	1.47235670	A011351	15 7	170859375
${\sqrt[{8}]{15}}$	1.40285055	A011352	15 8	2562890625
${\sqrt[{9}]{15}}$	1.35106675	A011353	15 9	38443359375
${\sqrt[{10}]{15}}$	1.31101942	A011354	15 10	576650390625
${\sqrt[{11}]{15}}$	1.27913795	A011355	15 11	8649755859375
${\sqrt[{12}]{15}}$	1.25316311	A011356	15 12	129746337890625
				A001024

Logarithms and fifteenth 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

(See A000584 for the fifth powers of integers).

Values for number theoretic functions with 15 as an argument

$\mu (15)$	1
$M(15)$	–1
$\pi (15)$	6
$\sigma _{1}(15)$	24
$\sigma _{0}(15)$	4
$\phi (15)$	8
$\Omega (15)$	2
$\omega (15)$	2
$\lambda (15)$	4	This is the Carmichael lambda function.
$\lambda (15)$	–1	This is the Liouville lambda function.
$\zeta (15)$	1.00003058823630702... (see A013673).
15!	1307674368000
$\Gamma (15)$	87178291200

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −15, 15

Neither ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ nor $\mathbb {Z} [{\sqrt {15}}]$ is a unique factorization domain. ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ only has two units, which makes things simpler and gives us more confidence in identifying instances of multiple distinct factorizations. But there are still traps for the unwary, such as the presence of so-called "half-integers."

For example, we could say that 16 has three distinct factorizations: 2 4, $(1-{\sqrt {-15}})(1+{\sqrt {-15}})$ and $\left({\frac {7}{2}}-{\frac {\sqrt {-15}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {-15}}{2}}\right)$. But that would be wrong. Each number in the latter two factorizations has nontrivial non-unit divisors and thus probably don't correspond to distinct factorizations any more than 4 2 does. First notice that the "half-integer" $\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)$ is fully an algebraic integer, with minimal polynomial $x^{2}-x+4$, and therefore an integer of this domain. Then verify that $2\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)=1+{\sqrt {-15}}$ and $\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)^{2}=-{\frac {7}{2}}+{\frac {\sqrt {-15}}{2}}$. With adjustments of signs, the other "composite" factors are addressed.

This raises the question of whether $2^{2}\left({\frac {1}{2}}-{\frac {\sqrt {-15}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)$ is a distinct factorization. If it is distinct, it is rather inelegant.

As for $\mathbb {Z} [{\sqrt {15}}]$, the of the MORE REMARKS GO HERE

$n$	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$	$\mathbb {Z} [{\sqrt {15}}]$
2	Irreducible
3
4	2 2 OR $\left({\frac {1}{2}}-{\frac {\sqrt {-15}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)$	2 2
5	Irreducible
6	2 × 3	2 × 3 OR $(-1)(3-{\sqrt {15}})(3+{\sqrt {15}})$
7	Prime	Irreducible
8	2 3
9	3 2
10	2 × 5 OR $\left({\frac {5}{2}}-{\frac {\sqrt {-15}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {-15}}{2}}\right)$	2 × 5 OR $(5-{\sqrt {15}})(5+{\sqrt {15}})$
11	Prime	$(-1)(2-{\sqrt {15}})(2+{\sqrt {15}})$
12	2 2 × 3
13	Prime
14	2 × 7	2 × 7 OR $(-1)(1-{\sqrt {15}})(1+{\sqrt {15}})$
15	3 × 5 OR $(-1)({\sqrt {-15}})^{2}$	3 × 5 OR $({\sqrt {15}})^{2}$
16	2 4 OR $\left({\frac {1}{2}}\pm {\frac {\sqrt {-15}}{2}}\right)^{2}$	2 4
17	Irreducible
18	2 × 3 2
19	$(2-{\sqrt {-15}})(2+{\sqrt {-15}})$	Prime
20	2 2 × 5

Ideals can help us make sense of the lack of unique factorization in these two rings.

$p$	Factorization of $\langle p\rangle$
	In ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$	In $\mathbb {Z} [{\sqrt {15}}]$
2	$\left\langle 2,{\frac {1}{2}}-{\frac {\sqrt {-15}}{2}}\right\rangle \left\langle 2,{\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right\rangle$	$\langle 2,1+{\sqrt {15}}\rangle ^{2}$
3	$\langle 3,{\sqrt {-15}}\rangle ^{2}$	$\langle 3,{\sqrt {15}}\rangle ^{2}$
5	$\langle 5,{\sqrt {-15}}\rangle ^{2}$	$\langle 5,{\sqrt {15}}\rangle ^{2}$
7	Prime	$\langle 7,1-{\sqrt {15}}\rangle \langle 7,1+{\sqrt {15}}\rangle$
11	Prime	$\langle 2-{\sqrt {15}}\rangle \langle 2+{\sqrt {15}}\rangle$
13	Prime
17	$\langle 17,6-{\sqrt {-15}}\rangle \langle 17,6+{\sqrt {-15}}\rangle$	$\langle 17,7-{\sqrt {15}}\rangle \langle 17,7+{\sqrt {15}}\rangle$
19	$\langle 2-{\sqrt {-15}}\rangle \langle 2+{\sqrt {-15}}\rangle$	Prime
23
29
31
37
41
43
47

Factorization of 15 in some quadratic integer rings

In $\mathbb {Z}$, 15 is composite, being the product of two primes. It is of course also composite in any quadratic integer rings, but it has different factorizations in some of those.

$\mathbb {Z} [i]$	$3(2\pm i)$
$\mathbb {Z} [{\sqrt {-2}}]$	$(1\pm {\sqrt {-2}})5$	$\mathbb {Z} [{\sqrt {2}}]$	3 × 5
$\mathbb {Z} [\omega ]$	$(-1)(1+2\omega )^{2}5$	$\mathbb {Z} [{\sqrt {3}}]$	$({\sqrt {3}})^{2}5$
$\mathbb {Z} [{\sqrt {-5}}]$	$(-1)3({\sqrt {-5}})^{2}$	$\mathbb {Z} [\phi ]$	$3(-1+2\phi )^{2}$
$\mathbb {Z} [{\sqrt {-6}}]$	3 × 5 OR $(3-{\sqrt {-6}})(3+{\sqrt {-6}})$	$\mathbb {Z} [{\sqrt {6}}]$	$(-1)(3\pm {\sqrt {6}})(1\pm {\sqrt {6}})$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$	3 × 5	$\mathbb {Z} [{\sqrt {7}}]$	$(-1)(2\pm {\sqrt {7}})5$
$\mathbb {Z} [{\sqrt {-10}}]$		$\mathbb {Z} [{\sqrt {10}}]$	3 × 5 OR $(5-{\sqrt {10}})(5+{\sqrt {10}})$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$	$\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)\left({\frac {3}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)$	$\mathbb {Z} [{\sqrt {11}}]$	$3(4\pm {\sqrt {11}})$
$\mathbb {Z} [{\sqrt {-13}}]$	3 × 5	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$	$(-1)\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)5$
$\mathbb {Z} [{\sqrt {-14}}]$		$\mathbb {Z} [{\sqrt {14}}]$	$(-1)3(3\pm {\sqrt {14}})$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$	3 × 5 OR $(-1)({\sqrt {-15}})^{2}$	$\mathbb {Z} [{\sqrt {15}}]$	3 × 5 OR $({\sqrt {15}})^{2}$
$\mathbb {Z} [{\sqrt {-17}}]$	3 × 5	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$	3 × 5
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$	$3\left({\frac {1}{2}}\pm {\frac {\sqrt {-19}}{2}}\right)$	$\mathbb {Z} [{\sqrt {19}}]$	$(-1)(4\pm {\sqrt {19}})(9\pm 2{\sqrt {19}})$

Representation of 15 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16 through 36
Representation	1111	120	33	30	23	21	17	16	15	14	13	12	11	10	F

We see that 15 is a Harshad number in bases 3, 5, 6, 7, 11, 12, 13 and trivially in base 15.

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