168 (number)

← 167 168 169 →
← 160 161 162 163 164 165 166 167 168 169 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred sixty-eight
Ordinal	168th (one hundred sixty-eighth)
Factorization	23 × 3 × 7
Divisors	1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
Greek numeral	ΡΞΗ´
Roman numeral	CLXVIII
Binary	101010002
Ternary	200203
Senary	4406
Octal	2508
Duodecimal	12012
Hexadecimal	A816

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.

Mathematics

Number theory

168 is the of fourth Dedekind number, ^[1] and one of sixty-five idoneal numbers. ^[2] It is one less then a square (13²), equal to the product of the first two perfect numbers ^[3]

${\displaystyle 168=6\times 28.}$

There are 168 primes less than 1000. ^{[lower-alpha 1]}

Composite index

The 128th composite number is 168, ^[4] one of a few numbers ${\displaystyle n}$ in the list of composites whose indices are the product of strings of digits of ${\displaystyle n}$ in decimal representation.

The first nine ${\displaystyle n}$ with this property are the following: ^[4]

• 48 as the 32nd (4 × 8 = 32)  ^{[lower-alpha 2]}
• 68 as the 48th (6 × 8 = 48)
• 78 is the 56th (7 × 8 = 56)
• 88 as the 64th (8² = 64)
• 98 as the 72nd (9 × 8 = 72)
• 128 as the 96th (12 × 8 = 96)
• 138 as the 104th (13 × 8 = 104)
• 158 as the 120th (15 × 8 = 120)
• 168 as the 128th (16 × 8 = 128)  ^{[lower-alpha 3]}

The next such number is 198 where 19 × 8 = 152 . The median between twenty-one integers [48, 68] is 58, where 148 is the median of forty-one integers [168, 128].

Totient values

For the Euler totient ${\displaystyle \varphi (n)}$ there is ${\displaystyle \varphi (168)=48}$, ^[5] where ${\displaystyle \varphi (48)=16}$ is also equivalent to the number of divisors of 168; ^[6] only eleven numbers have a totient of 48:{ 65 , 104, 105, 112, 130, 140, 144, 156, 168, 180, 210}. ^{[5] [lower-alpha 4]}

408, ^{[lower-alpha 5]} with a different permutation of the digits {0, 4, 8} where 048 is 48, has an totient of 128. So does the sum-of-divisors of 168, ^[9]

${\displaystyle \sigma (168)=480=2^{5}\times 3\times 5}$

as one of nine numbers total to have a totient of 128. ^[5]

Idoneal number

Leonard Euler noted 65 idoneal numbers (the most known, of only a maximum possible of two more), such that ${\displaystyle x^{2}+Dy^{2}}$ for an integer ${\displaystyle D}$, expressible in only one way, yields a prime power or twice a prime power. ^{[2] [10]}

Of these, 168 is the forty-fourth, where the smallest number to not be idoneal is the fifth prime number 11. ^[2] The largest such number 1848 (that is equivalent with the number of edges in the union of two cycle graphs of order 42) ^[11] contains a total of thirty-two divisors whose arithmetic mean is 180 ^{[12] [13]} (the second-largest number to have a totient of 48). ^[5] Preceding 1848 in the list of idoneal numbers is 1365, ^{[lower-alpha 6]} whose arithmetic mean of divisors is equal to 168 ^{[12] [13]} (while 1365 has a totient of 576 = 24²).

Where 48 is the 27th ideoneal number, 408 is the 58th. ^{[2] [lower-alpha 7]} On the other hand, the total count of known idoneal numbers (65), that is also equal to the sum of ten integers [2, ..., 11], has a sum-of-divisors of 84 (or, one-half of 168). ^[9]

Numbers of the form 2ⁿ

In base 10, 168 is the largest of ninety-two known ${\displaystyle n}$ such that ${\displaystyle 2^{n}}$ does not contain all numerical digits from that base (i.e. 0, 1, 2, ..., 9). ^[15]

${\displaystyle 2^{68}}$ is the first number to have such an expression where between the next two ${\displaystyle n}$ is an interval of ten integers: [70, 79]; ^[15] the median value between these is 74, the composite index of 100. ^{[4] [lower-alpha 8]}

Cunningham number

As a number of the form ${\displaystyle b^{n}\pm 1}$ for positive integers ${\displaystyle n}$, ${\displaystyle b}$ and ${\displaystyle b}$ not a perfect power, 168 is the thirty-second Cunningham number, ^[19] where it is one less than a square:

${\displaystyle 168=13^{2}-1.}$

On the other hand, 168 is one more than the third member of the fourth chain of nearly doubled primes of the first kind {41, 83, 167}, ^{[20] [21]} where 167 represents the thirty-ninth prime ^[22] (with 39 × 2 = 78). The smallest such chain is {2, 5, 11, 23, 47}.

Eisenstein series

168 is also coefficient four in the expansion of Eisenstein series ${\displaystyle E_{2}}$, ^[23] 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 ${\displaystyle A(n+1)=A(n)}$;^{cf. [4]} the latter holds a sum-of-divisors of 216 = 6 ³ , ^[9] which is the 168th composite number. ^[4]

Abstract algebra

168 is the number of maximal chains in the Bruhat order of symmetric group ${\displaystyle \mathrm {S} _{4},}$ ^[24] which is the largest solvable symmetric group with a total of ${\displaystyle 4!=24}$ elements.

168 is the order of the second smallest nonabelian simple group ${\displaystyle \mathrm {PSL} (2,7).}$ 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 ${\displaystyle \mathrm {PSL} (2,7)}$; ^[25] the Fano plane, isomorphic to the Klein group, has 168 symmetries.

In other fields

Dominoes

There are 168 pips on a double-six set of 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". ^[26]

Notes

1. (168, 1000) un-inclusive corresponds to a range of 831 integers, which is a value in equivalence with the composite index of 1000 = 10³. ^[4]
2. 32 is the twentieth composite.
3. 128 = 64 × 2 = 32 × 4, with 96 = 48 × 2, where also 168₁₀ = 120₁₂ (in duodecimal).
   On the other hand, 28 is the 18th composite number, ^[4]
4. The latter (210) is the 20th triangle number. ^[7]
5. 505, which is the magic constant of a ${\displaystyle 10\times 10}$ magic square, ^[8] is the 408th composite number.
6. 1365 ÷ 3 = 455 is the sum of (the first) ten terms in the sequence of numbers k ∈ {1, 2, 3, 4, 7, 8, 16, 31, 127, 256} such that k and k + 1 are prime powers. ^[14]
7. 840, with thirty-two divisors (the number with the largest number of divisors less than 1000), is the fourth-largest idoneal number. 88, 78, 58, 28, and 18 are also idoneal numbers, including 210 and 105 (numbers with totients of 48). ^[2]
8. In the iterative list of the A(n)-th composite number with A(1) = 11 where A(n + 1) = A(n), the first few elements are
   11, 20, 32, 48, 68, 93, 124, ... ^[16]
   which is preceded at 11 with the analogous list of successive super-primes ^[17] and primes ^[18] 11, 5, 3, 2, 1 (if the unit is a zeroth prime).
   The sum of these elements 1, 2, 3, 5, 11, 20, 32 is 74, with 32 + 68 = 100, and 48 in between.

References

1. Sloane, N. J. A. (ed.). "SequenceA000372(Dedekind numbers: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-01.
2. "Sloane's A000926 : Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28.
3. Sloane, N. J. A. (ed.). "SequenceA000396(Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-01.
4. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers: numbers n of the form x*y for x greater than 1 and y greater than 1.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-05.
5. Sloane, N. J. A. (ed.). "SequenceA000010(Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-05.
6. Sloane, N. J. A. (ed.). "SequenceA000005(d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-05.
7. Sloane, N. J. A. (ed.). "SequenceA000217(Triangular numbers: 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-09.
8. Sloane, N. J. A. (ed.). "SequenceA006003(a(n) as n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-08.
9. Sloane, N. J. A. (ed.). "SequenceA000203(The sum of divisors of n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-09.
10. Euler, Leonard (1806). "Illustratio paradoxi circa progressionem numerorum idoneorum sive congruorum". Nova Acta Academiae Scientarum Imperialis Petropolitinae. 15. Russian Academy of Sciences: 29–32. arXiv: math/0507352 . S2CID   118287274.
11. Sloane, N. J. A. (ed.). "SequenceA005563(a(n) as n*(n+2) equal to (n+1)^2 - 1.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-09.
12. Sloane, N. J. A. (ed.). "SequenceA003601(Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-16.
13. Sloane, N. J. A. (ed.). "SequenceA102187(Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-16.
14. Sloane, N. J. A. (ed.). "SequenceA006549(Numbers k such that k and k+1 are prime powers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-09.
15. "Sloane's A130696: Numbers k such that 2^k does not contain all ten decimal digits". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-19.
16. Sloane, N. J. A. (ed.). "SequenceA059407(a(n+1) as the a(n)-th composite number, with a(1) equal to 11.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-08.
17. Sloane, N. J. A. (ed.). "SequenceA006450(Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-08.
18. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-08.
19. Sloane, N. J. A. (ed.). "SequenceA080262(Cunningham numbers: of the form a^b +- 1, where a, b are greater than or equal to 2.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-20.
20. Sloane, N. J. A. (ed.). "SequenceA005602(Smallest prime beginning a complete Cunningham chain of length n (of the first kind).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-20.
21. Sloane, N. J. A. (ed.). "SequenceA348855(a(1) is 1. If a(n) is prime, a(n+1) is 2*a(n) + 1. If a(n) is not prime, a(n+1) is the least prime not already in the sequence.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-20.
22. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-20.
23. Sloane, N. J. A. (ed.). "SequenceA006352(Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-02.
24. Sloane, N. J. A. (ed.). "SequenceA061710(Number of maximal chains in the Bruhat order of S_n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-01.
25. "week214". math.ucr.edu. Retrieved 9 April 2023.
26. "$1 Prosperity Forever 168 Note - US Mint" . Retrieved 9 April 2023.

External links

• Number Facts and Trivia: 168
• The Number 168
• The Positive Integer 168
• Prime curiosities: 168