28

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28 is the second perfect number, the first to not be squarefree.

Membership in core sequences

Even numbers	..., 22, 24, 26, 28, 30, 32, 34, ...	A005843(14)
Composite numbers	..., 25, 26, 27, 28, 30, 32, 33, ...	A002808
Perfect numbers	6, 28, 496, 8128, 33550336, ...	A000396
Triangular numbers	..., 10, 15, 21, 28, 36, 45, 55, ...	A000217
Loeschian numbers	..., 21, 25, 27, 28, 31, 36, 37, ...	A003136

In Pascal's triangle, 28 occurs four times, the first two times on the eighth row as the sum of 7 and 21. In Lozanić's triangle, 28 also first appears in the eighth row, though as the sum of 12, 19 and a tacit −3 from the previous row.

Sequences pertaining to 28

Divisors of 28	1, 2, 4, 7, 14, 28	A018254
Multiples of 28	0, 28, 56, 84, 112, 140, 168, 196, ...	A135628
28-gonal numbers	1, 28, 81, 160, 265, 396, 553, 736, ...	A161935
Centered 28-gonal numbers	1, 29, 85, 169, 281, 421, 589, 785, ...	A195314
28-gonal pyramidal numbers	1, 29, 110, 270, 535, 931, 1484, ...	A256648
$3x+1$ sequence starting at 9	9, 28, 14, 7, 22, 11, 34, 17, 52, 26, ...	A033479
$3x-1$ sequence starting at 36	936, 18, 9, 26, 13, 38, 19, 56, 28, ...	A008894
$5x+1$ sequence starting at 11	11, 56, 28, 14, 7, 36, 18, 9, 46, 23, ...	A259193

Partitions of 28

There are 3718 partitions of 28. There are several nontrivial partitions of 28 into its own divisors, but since it is a perfect number, exactly only one of those consists of distinct divisors.

There are only two Goldbach representations of 28: 5 + 23 and 11 + 17.

Roots and powers of 28

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

$\sqrt{28}$	5.29150262	A010483	28 2	784
$\sqrt[3]{28}$	3.03658897	A010599	28 3	21952
$\sqrt[4]{28}$	2.30032663	A011023	28 4	614656
$\sqrt[5]{28}$	1.94729436	A011113	28 5	17210368
$\sqrt[6]{28}$	1.74258112		28 6	481890304
$\sqrt[7]{28}$	1.60967004		28 7	13492928512
$\sqrt[8]{28}$	1.51668277		28 8	377801998336
$\sqrt[9]{28}$	1.44808927		28 9	10578455953408
$\sqrt[10]{28}$	1.39545489		28 10	296196766695424
				A009972

Logarithms and 28th 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.

${\log }_{28}2$	0.20801459		${\log }_{2}28$	4.80735492		2 28	268435456
${\log }_{28}e$	0.30010162		$\log 28$	3.33220451	A016651	$e^{28}$	1446257064291.475173677
${\log }_{28}3$	0.32969533		${\log }_{3}28$	3.03310325		3 28	22876792454961
${\log }_{28}\pi$	0.34353530		${\log }_{\pi }28$	2.91090898		${\pi }^{28}$	83214007069229.61226
${\log }_{28}4$	0.41602919		${\log }_{4}28$	2.40367746		4 28	72057594037927936
${\log }_{28}5$	0.48299493		${\log }_{5}28$	2.07041507		5 28	37252902984619140625
${\log }_{28}6$	0.53770993		${\log }_{6}28$	1.85973874		6 28	6140942214464815497216
${\log }_{28}7$	0.58397080		${\log }_{7}28$	1.71241437		7 28	459986536544739960976801
${\log }_{28}8$	0.62404379		${\log }_{8}28$	1.60245164		8 28	19342813113834066795298816
${\log }_{28}9$	0.65939067		${\log }_{9}28$	1.51655162		9 28	523347633027360537213511521
${\log }_{28}10$	0.69100953		${\log }_{10}28$	1.44715803		10 28	10000000000000000000000000000

See A122969 for the 28th powers of integers.

Values for number theoretic functions with 28 as an argument

$\mu (28)$	0
$M(28)$	–1
$\pi (28)$	9
${\sigma }_1(28)$	56	Note that this is twice 28.
${\sigma }_0(28)$	6
$\varphi (28)$	12
$\mathrm{\Omega }(28)$	3
$\omega (28)$	2
$\lambda (28)$	6	This is the Carmichael lambda function.
$\lambda (28)$	–1	This is the Liouville lambda function.
$\zeta (28)$	1.0000000037253340247884570548192...
28!	304888344611713860501504000000
$\mathrm{\Gamma }(28)$	10888869450418352160768000000

Factorization of 28 in some quadratic integer rings

As was mentioned above, the prime factorization of 28 is $2^2\times 7$ in $\mathbb{\mathbb{Z}}$. But it has different factorizations in some quadratic integer rings.

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

Representation of 28 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	11100	1001	130	103	44	40	34	31	28	26	24	22	20	1D	1C	1B	1A	19	18

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