62 (number)

← 61 62 63 →
← 60 61 62 63 64 65 66 67 68 69 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	sixty-two
Ordinal	62nd (sixty-second)
Factorization	2 × 31
Divisors	1, 2, 31, 62
Greek numeral	ΞΒ´
Roman numeral	LXII, lxii
Binary	1111102
Ternary	20223
Senary	1426
Octal	768
Duodecimal	5212
Hexadecimal	3E16

62 (sixty-two) is the natural number following 61 and preceding 63.

In mathematics

62 as the sum of three distinct positive squares.

62 is:

• the eighteenth discrete semiprime (${\displaystyle 2\times 31}$) and tenth of the form (2.q), where q is a higher prime.
• with an aliquot sum of 34; itself a semiprime, within an aliquot sequence of seven composite numbers (62,34,20,22,14,10,8,7,1,0) to the Prime in the 7-aliquot tree. This is the longest aliquot sequence for a semiprime up to 118 which has one more sequence member. 62 is the tenth member of the 7-aliquot tree (7, 8, 10, 14, 20, 22, 34, 38, 49, 62, 75, 118, 148, etc).
• a nontotient. ^[1]
• palindromic and a repdigit in bases 5 (222₅) and 30 (22₃₀)
• the sum of the number of faces, edges and vertices of icosahedron or dodecahedron.
• the number of faces of two of the Archimedean solids, the rhombicosidodecahedron and truncated icosidodecahedron.
• the smallest number that is the sum of three distinct positive squares in two (or more) ways, ${\displaystyle 1^{2}+5^{2}+6^{2}=2^{2}+3^{2}+7^{2}}$ ^[2]
• the only number whose cube in base 10 (238328) consists of 3 digits each occurring 2 times. ^[3]
• The 20th & 21st, 72nd & 73rd, 75th & 76th digits of pi. ^[4]

Square root of 62

As a consequence of the mathematical coincidence that 10⁶ − 2 = 999,998 = 62 × 127², the decimal representation of the square root of 62 has a curiosity in its digits: ^[5]

${\displaystyle {\sqrt {62}}}$ = 7.874 007874 011811 019685 034448 812007 …

For the first 22 significant figures, each six-digit block is 7,874 or a half-integer multiple of it.

7,874 × 1.5 = 11,811

7,874 × 2.5 = 19,685

The pattern follows from the following polynomial series:

$${\displaystyle {\begin{aligned}(1-2x)^{-{\frac {1}{2}}}&=1+x+{\frac {3}{2}}x^{2}+{\frac {5}{2}}x^{3}+{\frac {35}{8}}x^{4}+{\frac {63}{8}}x^{5}+\cdots \end{aligned}}}$$

Plugging in x = 10⁻⁶ yields ${\displaystyle {\frac {1}{\sqrt {999,998}}}}$, and ${\displaystyle {\sqrt {62}}}$ = ${\displaystyle {7,874}\times {\frac {1}{\sqrt {999,998}}}}$.

References

1. "Sloane's A005277 : Nontotients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
2. "A024804: Numbers that are the sum of 3 distinct nonzero squares in 2 or more ways". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-03-25.
3. John D. Cook (5 February 2010). "Carnival of Mathematics #62".
4. "On the Number 62". www.wisdomportal.com. Retrieved 2021-01-21.
5. Robert Munafo. "Notable Properties of Specific Numbers: 62".