7

This article is under construction.            

Please do not rely on any information it contains.            

7 is a prime number.

Membership in core sequences

Odd numbers	1, 3, 5, 7, 9, 11, 13, ...	A005408
Prime numbers	2, 3, 5, 7, 11, 13, 17, ...	A000040
Partition numbers	1, 1, 2, 3, 5, 7, 11, 15, ...	A000041
Lucas numbers	2, 1, 3, 4, 7, 11, 18, 29, ...	A000032

In Pascal's triangle, 7 occurs twice, corresponding to $(\genfrac{}{}{0ex}{}{7}{1})$ and $(\genfrac{}{}{0ex}{}{7}{6})$. (In Lozanić's triangle, 7 occurs four times).

Sequences pertaining to 7

Multiples of 7	0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, ...	A008585
Inert rational primes in $\mathbb{\mathbb{Q}}(\sqrt{7})$	5, 11, 13, 17, 23, 41, 43, 61, 67, 71, 73, 79, 89, ...	A003630
Heptagonal numbers	0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, ...	A000566
Centered heptagonal numbers	1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, ...	A069099
$3x+1$ sequence beginning at 9	9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, ...	A033479

Partitions of 7

There are fifteen partitions of 7, of which five consist of distinct numbers: {1, 2, 4}, {1, 6}, {2, 5}, {3, 4} and {7}. There are three partitions of 7 into primes, and those are {7}, {2, 5}, {2, 2, 3}.

Roots and powers of 7

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

$\sqrt{7}$	2.64575131	A002163	7 2	49
$\sqrt[3]{7}$	1.91293118	A005482	7 3	343
$\sqrt[4]{7}$	1.62657656	A011005	7 4	2401
$\sqrt[5]{7}$	1.47577316	A011092	7 5	16807
$\sqrt[6]{7}$	1.38308755	A011230	7 6	117649
$\sqrt[7]{7}$	1.32046924	A011231	7 7	823543
$\sqrt[8]{7}$	1.27537310	A011232	7 8	5764801
$\sqrt[9]{7}$	1.24136581	A011233	7 9	40353607
$\sqrt[10]{7}$	1.21481404	A011234	7 10	282475249
$\sqrt[11]{7}$	1.19351283	A011235	7 11	1977326743
$\sqrt[12]{7}$	1.17604742	A011236	7 12	13841287201
				A000420

Logarithms and seventh 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.

${\log }_{7}2$	0.35620718	A152713	${\log }_{2}7$	2.80735492	A020860	2 7	128
${\log }_{7}e$	0.51389834		$\log 7$	1.94591014	A016630	$e^7$	1096.63315842	A092513
${\log }_{7}3$	0.56457503	A152945	${\log }_{3}7$	1.77124374	A152565	3 7	2187
${\log }_7\pi$	0.58827479		${\log }_{\pi }7$	1.69988586		${\pi }^7$	3020.29322777	A092735
${\log }_{7}4$	0.71241437	A153103	${\log }_{4}7$	1.40367746	A153615	4 7	16384
${\log }_{7}5$	0.82708747	A153203	${\log }_{5}7$	1.20906195	A153616	5 7	78125
${\log }_{7}6$	0.92078222	A153463	${\log }_{6}7$	1.08603313	A153617	6 7	279936
${\log }_{7}7$			1.00000000			7 7	823543
${\log }_{7}8$	1.06862156	A153755	${\log }_{8}7$	0.93578497	A153618	8 7	2097152
${\log }_{7}9$	1.12915006	A113211	${\log }_{9}7$	0.88562187	A153619	9 7	4782969
${\log }_{7}10$	1.18329466	A154158	${\log }_{10}7$	0.84509804	A153620	10 7	10000000

(See A001015 for the seventh powers of integers).

Values for number theoretic functions with 7 as an argument

$\mu (7)$	–1
$M(7)$	–2
$\pi (7)$	4
${\sigma }_1(7)$	8
${\sigma }_0(7)$	2
$\varphi (7)$	6
$\mathrm{\Omega }(7)$	1
$\omega (7)$	1
$\lambda (7)$	6	This is the Carmichael lambda function.
$\lambda (7)$	–1	This is the Liouville lambda function.
$\zeta (7)$	1.0083492773819228268397975498... (see A013665).
7!	5040
$\mathrm{\Gamma }(7)$	24

Factorization of some small integers in a quadratic integer ring adjoining the square root of −7 or 7

Both ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$ and $\mathbb{\mathbb{Z}}[\sqrt{7}]$ are unique factorization domains. But ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$ includes algebraic integers of the form $(\frac{a}{2}+\frac{b\sqrt{-7}}{2})$ (with $a$ and $b$ both odd) and has only two units (1 and –1) whereas $\mathbb{\mathbb{Z}}[\sqrt{7}]$ has infinitely many units and no "half-integers."

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

Factorization of 7 in some quadratic integer rings

In $\mathbb{\mathbb{Z}}$, 7 is a prime number. But it has different factorizations in some quadratic integer rings.

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

Representation of 7 in various bases

Base	2	3	4	5	6	7	8 through 36
Representation	111	21	13	12	11	10	7

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