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 fourth Dedekind number, ^[1] and one of sixty-five idoneal numbers. ^[2] It is one less than 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 and sigma 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]}

The number of divisors of 168 is 16, ^[8] making it a largely composite number. ^[9]

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, ^[11]

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

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

48 sets the sixteenth record for sum-of-divisors of positive integers (of 124), and the seventeenth record value is 168, ^[12] from six numbers (60, 78, 92, 123, 143, and 167). ^[11]

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 ${\displaystyle x^{2}+Dy^{2}}$ for an integer ${\displaystyle D}$, expressible in only one way, yields a prime power or twice a prime power. ^{[2] [13]}

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) ^[14] contains a total of thirty-two divisors whose arithmetic mean is 180 ^{[15] [16]} (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 ^{[15] [16]} (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). ^[11]

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). ^[18]

${\displaystyle 2^{68}}$ is the first number to have such an expression where between the next two ${\displaystyle n}$ is an interval of ten integers: [70, 79]; ^[18] the median values between these are (75, 74), where the smaller of these two values represents 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, ^[22] 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}, ^{[23] [24]} where 167 represents the thirty-ninth prime ^[25] (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 ${\displaystyle E_{2}}$, ^[26] 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 , ^[11] which is the 168th composite number. ^[4]

Abstract algebra

168 is the number of maximal chains in the Bruhat order of symmetric group ${\displaystyle \mathrm {S} _{4},}$ ^[27] 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 ${\displaystyle \mathrm {PSL} (2,7)}$; ^[28] 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 一六八 ("168") Mandarin pinyin :yīliùbā is roughly homophonous with the phrase "一路發" Mandarin pinyin :yīlùfā which means "fortune all the way", or, as the United States Mint claims, "Prosperity Forever". ^[29]

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, ^[10] 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. ^[17]
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, ... ^[19]
   which is preceded at 11 with the analogous list of successive super-primes ^[20] and primes ^[21] 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.). "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.
9. Sloane, N. J. A. (ed.). "SequenceA067128(Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
10. 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.
11. 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.
12. Sloane, N. J. A. (ed.). "SequenceA034885(Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-07-06.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. "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.
19. 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.
20. 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.
21. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-08.
22. 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.
23. 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.
24. 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.
25. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-20.
26. 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.
27. 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.
28. "week214". math.ucr.edu. Retrieved 9 April 2023.
29. "$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