This site is supported by donations to The OEIS Foundation.

7

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 ${\displaystyle {\binom {7}{1}}}$ and ${\displaystyle {\binom {7}{6}}}$. (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 ${\displaystyle \mathbb {Q} ({\sqrt {7}})}$ 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 ${\displaystyle 3x+1}$ 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.

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

Logarithms and seventh powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.

As above, irrational numbers in the following table are truncated to eight decimal places.

 ${\displaystyle \log _{7}2}$ 0.35620718 A152713 ${\displaystyle \log _{2}7}$ 2.80735492 A020860 2 7 128 ${\displaystyle \log _{7}e}$ 0.51389834 ${\displaystyle \log 7}$ 1.94591014 A016630 ${\displaystyle e^{7}}$ 1096.63 A092513 ${\displaystyle \log _{7}3}$ 0.56457503 A152945 ${\displaystyle \log _{3}7}$ 1.77124374 A152565 3 7 2187 ${\displaystyle \log _{7}\pi }$ 0.58827479 ${\displaystyle \log _{\pi }7}$ 1.69988586 ${\displaystyle \pi ^{7}}$ 3020.29 A092735 ${\displaystyle \log _{7}4}$ 0.71241437 A153103 ${\displaystyle \log _{4}7}$ 1.40367746 A153615 4 7 16384 ${\displaystyle \log _{7}5}$ 0.82708747 A153203 ${\displaystyle \log _{5}7}$ 1.20906195 A153616 5 7 78125 ${\displaystyle \log _{7}6}$ 0.92078222 A153463 ${\displaystyle \log _{6}7}$ 1.08603313 A153617 6 7 279936 ${\displaystyle \log _{7}7}$ 1.00000000 7 7 823543 ${\displaystyle \log _{7}8}$ 1.06862156 A153755 ${\displaystyle \log _{8}7}$ 0.93578497 A153618 8 7 2.09715e+06 ${\displaystyle \log _{7}9}$ 1.12915006 A113211 ${\displaystyle \log _{9}7}$ 0.88562187 A153619 9 7 4.78297e+06 ${\displaystyle \log _{7}10}$ 1.18329466 A154158 ${\displaystyle \log _{10}7}$ 0.84509804 A153620 10 7 1e+07

(See A001015 for the seventh powers of integers).

Values for number theoretic functions with 7 as an argument

 ${\displaystyle \mu (7)}$ –1 ${\displaystyle M(7)}$ –2 ${\displaystyle \pi (7)}$ 4 ${\displaystyle \sigma _{1}(7)}$ 8 ${\displaystyle \sigma _{0}(7)}$ 2 ${\displaystyle \phi (7)}$ 6 ${\displaystyle \Omega (7)}$ 1 ${\displaystyle \omega (7)}$ 1 ${\displaystyle \lambda (7)}$ 6 This is the Carmichael lambda function. ${\displaystyle \lambda (7)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (7)}$ 1.0083492773819228268397975498... (see A013665). 7! 5040 ${\displaystyle \Gamma (7)}$ 24

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

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

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

Factorization of 7 in some quadratic integer rings

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

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

Representation of 7 in various bases

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

 ${\displaystyle -1}$ 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