666

Number
$666$ (six hundred and sixty-six) is:


 * $2 \times 3^2 \times 37$


 * The $3$rd and last after $55$, $66$ of the $3$ repdigit numbers which are also triangular.


 * The total of all the entries in a magic square of order $6$, after $1$, $(10)$, $45$, $136$, $325$:
 * $666 = \ds \sum_{k \mathop = 1}^{6^2} k = \dfrac {6^2 \paren {6^2 + 1} } 2$


 * The $7$th after $4$, $13$, $38$, $87$, $208$, $377$ in the sequence formed by adding the squares of the first $n$ primes:
 * $666 = \ds \sum_{i \mathop = 1}^7 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2$


 * The $8$th positive integer after $1$, $24$, $26$, $87$, $168$, $388$, $594$ whose Euler $\phi$ value is equal to the product of its digits:
 * $\map \phi {666} = 216 = 6 \times 6 \times 6$


 * The $9$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$


 * The $34$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$:
 * $6 + 6 + 6 = 2 + 3 + 3 + 3 + 7 = 18$


 * The $36$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$:
 * $666 = \ds \sum_{k \mathop = 1}^{36} k = \dfrac {36 \times \paren {36 + 1} } 2$


 * The basis of an approximation to the Golden Ratio correct to $10$ decimal places:
 * $\phi \approx -2 \map \sin {666} = -1.61803 \, 39887 \, 5 \ldots$