65 (number)

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

65 is the nineteenth distinct semiprime,^[1] (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 number (65,19,1,0) to the prime; as the first member' of the 19-aliquot tree.

It is an octagonal number.^[2] It is also a Cullen number.^[3] Given 65, the Mertens function returns 0.^[4]

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

${\displaystyle {\begin{bmatrix}17&24&1&8&15\\23&5&7&14&16\\4&6&13&20&22\\10&12&19&21&3\\11&18&25&2&9\end{bmatrix}}.}$

This number is also the magic constant of n-Queens Problem for n = 5.^[5]

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.^[6]

It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).^[7]

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.^[8]

65 = 1⁵ + 2⁴ + 3³ + 4² + 5¹.^[9]

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.^[10] The first two are "primitive", and 65 is the lowest number to be the largest side of more than one such triple.^[11]

65 is the number of compositions of 11 into distinct parts.^[12]

• 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.