62 (number)

In mathematics

62 is:

• the eighteenth discrete semiprime (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.
• 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, 1^2+5^2+6^2 = 2^2+3^2+7^2
• the only number whose cube in base 10 (238328) consists of 3 digits each occurring 2 times.{{cite web
• The 20th and 21st, the 72nd and 73rd, and the 75th and 76th digits of .

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:

\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:

\begin{align}

(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{align}

Plugging in x = 10−6 yields \frac1{\sqrt{999,998}}, and \sqrt{62} = {7,874} \times \frac1{\sqrt{999,998}}.

References

References

1. "Sloane's A005277 : Nontotients". OEIS Foundation.
2. "A024804: Numbers that are the sum of 3 distinct nonzero squares in 2 or more ways". OEIS Foundation.
3. "On the Number 62".
4. Robert Munafo. "Notable Properties of Specific Numbers: 62".