[mickey]

[P. De Geest and G. L. Honaker, Jr.] Now that we have the Sk(n) notation, define SP(n) as the sum of the first n palindromic primes. Then:

S3( SP(666) ) = 3 · 666

where the same digits (3, 666) appear on both sides of the equation!
[by Carlos Rivera] The number 20772199 is the smallest integer with the property that the sum of the prime factors of n and the sum of the prime factors of n+1 are both equal to 666:

20772199 = 7 x 41 x 157 x 461, and 7+41+157+461 = 666
20772200 = 2x2x2x5x5x283x367, and 2+2+2+5+5+283+367 = 666.

Of course, integers n and n+1 having the same sum of prime factors are the famous Ruth-Aaron pairs. So we can say that (20772199, 20772200) is the smallest beastly Ruth-Aaron pair.
[by G. L. Honaker, Jr.] The sum of the first 666 primes contains 666:

2 + 3 + 5 + 7 + 11 · · · + 4969 + 4973 = 1533157 = 23 · 66659

[Wang, J. Rec. Math, 26(3)] The number 666 is related to the golden ratio! (If a rectangle has the property that cutting off a square from it leaves a rectangle whose proportions are the same as the original, then that rectangle's proportions are in the golden ratio. Also, the golden ratio is the limit, as n becomes large, of the ratio between adjacent numbers in the Fibonacci sequence.) Denoting the Golden Ratio by t, we have the following identity, where the angles are in degrees:

sin(666) = cos(6·6·6) = -t/2

which can be combined into the lovely expression:

t = - (sin(666) + cos(6·6·6) )

There are exactly two ways to insert '+' signs into the sequence 123456789 to make the sum 666, and exactly one way for the sequence 987654321:

666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9
666 = 9 + 87 + 6 + 543 + 21

[from Patrick Capelle]

666 is a divisor of 123456789 + 987654321.

[from Patrick Capelle] 666 can be expressed just using the digit 3, as

666 = p(3·3·3) + p(p(3·3·3))

where p(i) means the ith prime number (that is, p(1) = 2, p(2) = 3, p(3) = 5, etc.)
A Smith number is an integer in which the sum of its digits is equal to the sum of the digits of its prime factors. 666 is a Smith number, since

666 = 2·3·3·37

while at the same time

6 + 6 + 6 = 2 + 3 + 3 + 3 + 7.

Consider integers n with the following special property: if n is written in binary, then the one's complement is taken (which changes all 1's to 0's and all 0's to 1's), then the result is written in reverse, the result is the starting integer n. The first few such numbers are

2 10 12 38 42 52 56 142 150 170 178 204 212 232 240 542 558 598 614...

For example, 38 is 100110, which complemented is 011001, which reversed is 100110. Now, you don't really need to be told what the next one after 614 is, do you?
The following fact is quite well known, but still interesting: If you write the first 6 Roman numerals, in order from largest to smallest, you get 666:

DCLXVI = 666.

The previous one suggests a form of word play that was popular several centuries ago: the chronogram. A chronogram attaches a numerical value to an English phrase or sentence by summing up the values of any Roman numerals it contains. (Back then, U,V and I,J were often considered the same letter for the purpose of the chronogram, however I prefer to distinguish them.) What's the best English chronogram for 666? Here's one that's pretty apt:

Expect The Devil.

Note that four of the six numerals are contained in the last word.
A standard function in number theory is phi(n), which is the number of integers smaller than n and relatively prime to n. Remarkably,

phi(666) = 6·6·6.

The nth triangular number is given by the formula T(n) = (n)(n+1)/2, and is equal to the sum of the numbers from 1 to n.
666 is the 36th triangular number - in other words,

T(6·6) = 666.

In 1975 Ballew and Weger proved (see J. Rec. Math, Vol. 8, No. 2

666 is the largest triangular number that's also a repdigit

(A repdigit is a number consisting of a single repeated non-zero digit, like 11 or 22 or 555555.)

III