65 (number)

Template:Infobox number 65 (sixty-five) is the natural number following 64 and preceding 66.

In mathematics

File:Square-sum-65.png

65 as the sum of distinct positive squares.

65 is the nineteenth distinct semiprime,<ref>Template:Cite OEIS</ref> (5.13); and the third of the form (5.q), where q is a higher prime.

65 has a prime aliquot sum of 19 within an aliquot sequence of one composite numbers (65,19,1,0) to the prime; as the first member' of the 19-aliquot tree.

It is an octagonal number.<ref>Template:Cite web</ref> It is also a Cullen number.<ref>Template:Cite web</ref> Given 65, the Mertens function returns 0.<ref>Template:Cite web</ref>

This number is the magic constant of a 5×5 normal magic square:

<math> \begin{bmatrix}

17 & 24 & 1 & 8 & 15 \\
23 & 5 & 7 & 14 & 16 \\
4 & 6 & 13 & 20 & 22 \\
10 & 12 & 19 & 21 & 3 \\
11 & 18 & 25 & 2 & 9

\end{bmatrix}. </math>

This number is also the magic constant of n-Queens Problem for n = 5.<ref>Template:Cite OEIS</ref>

65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two (or more) ways, 65 = 8² + 1² = 7² + 4².<ref>Template:OEIS2C</ref>

It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).<ref>Template:Cite web</ref>

65 is a Stirling number of the second kind, the number of ways of dividing a set of six objects into four non-empty subsets.<ref>Template:Cite web</ref>

65 = 1⁵ + 2⁴ + 3³ + 4² + 5¹.<ref>Template:OEIS2C</ref>

65 is the length of the hypotenuse of 4 different Pythagorean triangles, the lowest number to have more than 2: 65² = 16² + 63² = 33² + 56² = 39² + 52² = 25² + 60².<ref>Template:OEIS2C</ref> The first two are "primitive", and 65 is the lowest number to be the largest side of more than one such triple.<ref>Template:OEIS2C</ref>

65 is the number of compositions of 11 into distinct parts.<ref>Template:Cite OEIS</ref>

In other fields

• the traditional age for retirement in the United Kingdom, Germany, the United States, Canada, and several other countries.
• A 65th anniversary is sometimes referred to as a sapphire jubilee.