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# 77

Please do not rely on any information it contains.

77 is an integer, the largest number that can't be written as a sum of distinct numbers whose reciprocals sum to 1.

## Membership in core sequences

 Odd numbers ..., 71, 73, 75, 77, 79, 81, 83, ... A005843 Composite numbers ..., 74, 75, 76, 77, 78, 80, 81, ... A002808 Semiprimes ..., 65, 69, 74, 77, 82, 85, 86, ... A001358 Squarefree numbers ..., 71, 73, 74, 77, 78, 79, 82, ... A005117

## Sequences pertaining to 77

 Multiples of 77 77, 154, 231, 308, 385, 462, 539, 616, 693, 770, 847, 924, ... $3x+1$ sequence beginning at 51 51, 154, 77, 232, 116, 58, 29, 88, 44, 22, 11, 34, 17, 52, ... A008883

## Partitions of 77

There are 10619863 partitions of 77. Of these, the of the [FINISH WRITING]

## Roots and powers of 77

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

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## Factorization of some small integers in a quadratic integer ring adjoining ${\sqrt {-77}}$ , ${\sqrt {77}}$ REMARKS GO HERE

 $n$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {77}})}$ 2 Prime 3 4 2 2 5 Prime 6 2 × 3 7 $(-1)\left({\frac {7}{2}}-{\frac {\sqrt {77}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {77}}{2}}\right)$ 8 2 3 9 3 2 10 2 × 5 11 $\left({\frac {11}{2}}-{\frac {\sqrt {77}}{2}}\right)\left({\frac {11}{2}}+{\frac {\sqrt {77}}{2}}\right)$ 12 2 2 × 3 13 $(-1)\left({\frac {5}{2}}-{\frac {\sqrt {77}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {77}}{2}}\right)$ 14 $(-1)2\left({\frac {7}{2}}\pm {\frac {\sqrt {77}}{2}}\right)$ 15 3 × 5 16 2 4 17 $(-1)\left({\frac {3}{2}}-{\frac {\sqrt {77}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {77}}{2}}\right)$ 18 2 × 3 2 19 $(-1)\left({\frac {1}{2}}-{\frac {\sqrt {77}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {77}}{2}}\right)$ 20 2 2 × 5

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PLACEHOLDER

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## Representation of 77 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1001101 2212 1031 302 205 140 115 85 77 70 65 5C 57 52 4D 49 45 41 3H

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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