168 (number)

168 (one hundred [and] sixty-eight) is the natural number following 167 and preceding 169.

It is the number of hours in a week, or 7 x 24 hours.

Number theory
168 is the fourth Dedekind number, and one of sixty-five idoneal numbers. It is one less than a square (132), equal to the product of the first two perfect numbers


 * $$168 = 6 \times 28.$$

There are 168 primes less than 1000.

Composite index
The 128th composite number is 168, one of a few numbers $$n$$ in the list of composites whose indices are the product of strings of digits of $$n$$ in decimal representation.

The first nine $$n$$ with this property are the following:

The next such number is 198 where $(168, 1000)$. The median between twenty-one integers $(4 × 8 = 32)$ is 58, where 148 is the median of forty-one integers $(6 × 8 = 48)$.

Totient and sigma values
For the Euler totient $$\varphi (n)$$ there is $$\varphi (168) = 48$$, where $$\varphi (48) = 16$$ is also equivalent to the number of divisors of 168; only eleven numbers have a totient of 48:$(7 × 8 = 56)$.

408, with a different permutation of the digits $(8^{2} = 64)$ where 048 is 48, has an totient of 128. So does the sum-of-divisors of 168,
 * $$\sigma (168) = 480 = 2^{5} \times 3 \times 5$$

as one of nine numbers total to have a totient of 128.

48 sets the sixteenth record for sum-of-divisors of positive integers (of 124), and the seventeenth record value is 168, from six numbers (60, 78, 92, 123, 143, and 167).

The difference between 168 and 48 is the factorial of five (120), where their sum is the cube of six (216).

Idoneal number
Leonhard Euler noted 65 idoneal numbers (the most known, of only a maximum possible of two more), such that $$x^{2} +Dy^{2}$$ for an integer $$D$$, expressible in only one way, yields a prime power or twice a prime power.

Of these, 168 is the forty-fourth, where the smallest number to not be idoneal is the fifth prime number 11. The largest such number 1848 (that is equivalent with the number of edges in the union of two cycle graphs of order 42) contains a total of thirty-two divisors whose arithmetic mean is 180 (the second-largest number to have a totient of 48). Preceding 1848 in the list of idoneal numbers is 1365, whose arithmetic mean of divisors is equal to 168 (while 1365 has a totient of 576 = 242).

Where 48 is the 27th ideoneal number, 408 is the 58th. On the other hand, the total count of known idoneal numbers (65), that is also equal to the sum of ten integers $(9 × 8 = 72)$, has a sum-of-divisors of 84 (or, one-half of 168).

Numbers of the form 2n
In base 10, 168 is the largest of ninety-two known $$n$$ such that $$2^{n}$$ does not contain all numerical digits from that base (i.e. 0, 1, 2, ..., 9).

$$2^{68}$$ is the first number to have such an expression where between the next two $$n$$ is an interval of ten integers: $(12 × 8 = 96)$; the median values between these are (75, 74), where the smaller of these two values represents the composite index of 100.

Cunningham number
As a number of the form $$b^n\pm1$$ for positive integers $$n$$, $$b$$ and $$b$$ not a perfect power, 168 is the thirty-second Cunningham number, where it is one less than a square:
 * $$168 = 13^{2} - 1.$$

On the other hand, 168 is one more than the third member of the fourth chain of nearly doubled primes of the first kind $(13 × 8 = 104)$, where 167 represents the thirty-ninth prime (with 39 × 2 = 78). The smallest such chain is $(15 × 8 = 120)$.

Eisenstein series
168 is also coefficient four in the expansion of Eisenstein series $$E_{2}$$, which also includes 144 and 96 (or 48 × 2) as the fifth and third coefficients, respectively — these have a sum of 240, which follows 144 and 187 in the list of successive composites $$A(n + 1) = A(n)$$;cf. the latter holds a sum-of-divisors of 216 = 63, which is the 168th composite number.

Abstract algebra
168 is the number of maximal chains in the Bruhat order of symmetric group $$\mathrm {S}_4,$$ which is the largest solvable symmetric group with a total of $$4! = 24$$ elements.

168 is the order of the second smallest nonabelian simple group $$\mathrm {PSL}(2,7).$$ From Hurwitz's automorphisms theorem, 168 is the maximum possible number of automorphisms of a genus 3 Riemann surface, this maximum being achieved by the Klein quartic, whose symmetry group is $$\mathrm {PSL}(2,7)$$; the Fano plane, isomorphic to the Klein group, has 168 symmetries.

Dominoes


In the game of dominoes, tiles are marked with a number of spots, or pips. A Double 6 set of 28 tiles contains a total of 168 pips.

Numerology
Some Chinese consider 168 a lucky number, because it is roughly homophonous with the phrase "一路發" which means "fortune all the way", or, as the United States Mint claims, "Prosperity Forever".