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# 37

Please do not rely on any information it contains.

37 is is the maximum number of fifth powers needed to sum to any number.

## Membership in core sequences

 Odd numbers ..., 31, 33, 35, 37, 39, 41, 43, ... A005408 Prime numbers ..., 23, 29, 31, 37, 41, 43, 47, ... A000040 Squarefree numbers ..., 33, 34, 35, 37, 38, 39, 41, ... A005117

In Pascal's triangle, 37 occurs twice.

## Sequences pertaining to 37

 Multiples of 37 0, 37, 74, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444, ... A085959 $3x+1$ sequence starting at 87 87, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, ... A008879

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## Values for number theoretic functions with 37 as an argument

 $\mu (37)$ –1 $M(37)$ –2 $\pi (37)$ 11 $\sigma _{1}(37)$ 38 $\sigma _{0}(37)$ 2 $\phi (37)$ 36 $\Omega (37)$ 1 $\omega (37)$ 1 $\lambda (37)$ 36 This is the Carmichael lambda function. $\lambda (37)$ –1 This is the Liouville lambda function. $\zeta (37)$ 37! 13763753091226345046315979581580902400000000 $\Gamma (37)$ 371993326789901217467999448150835200000000

## Factorization of some small integers in a quadratic integer ring adjoining ${\sqrt {-37}}$ , ${\sqrt {37}}$ The commutative quadratic integer ring with unity ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {37}})}$ , with units of the form $\pm (6+{\sqrt {37}})^{n}\,$ ($n\in \mathbb {Z}$ ), is a unique factorization domain, and it is norm-Euclidean.

 $n$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {37}})}$ 2 Prime 3 $\left({\frac {7}{2}}-{\frac {\sqrt {37}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {37}}{2}}\right)$ 4 2 2 5 Prime 6 $2\left({\frac {7}{2}}-{\frac {\sqrt {37}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {37}}{2}}\right)$ 7 $\left({\frac {55}{2}}-{\frac {9{\sqrt {37}}}{2}}\right)\left({\frac {55}{2}}+{\frac {9{\sqrt {37}}}{2}}\right)$ 8 2 3 9 $\left({\frac {7}{2}}-{\frac {\sqrt {37}}{2}}\right)^{2}\left({\frac {7}{2}}+{\frac {\sqrt {37}}{2}}\right)^{2}$ 10 2 × 5 11 $(-1)\left({\frac {17}{2}}-{\frac {3{\sqrt {37}}}{2}}\right)\left({\frac {17}{2}}+{\frac {3{\sqrt {37}}}{2}}\right)$ 12 $2^{2}\left({\frac {7}{2}}-{\frac {\sqrt {37}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {37}}{2}}\right)$ 13 Prime 14 $2\left({\frac {55}{2}}-{\frac {9{\sqrt {37}}}{2}}\right)\left({\frac {55}{2}}+{\frac {9{\sqrt {37}}}{2}}\right)$ 15 $3\left({\frac {7}{2}}-{\frac {\sqrt {37}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {37}}{2}}\right)$ 16 2 4 17 Prime 18 $2\left({\frac {7}{2}}-{\frac {\sqrt {37}}{2}}\right)^{2}\left({\frac {7}{2}}+{\frac {\sqrt {37}}{2}}\right)^{2}$ 19 Prime 20 2 2 × 5

Unlike ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {37}})}$ , $\mathbb {Z} [{\sqrt {-37}}]$ is not a unique factorization domain. But the window of 2 through 21 does not provide as interesting a window for the of the [FINISH WRITING]

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## Representation of 37 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 100101 1101 211 122 101 52 45 41 37 34 31 2B 29 27 25 23 21 1I 1H

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 $-1$ 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