168 (number)
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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
[edit]Number theory
[edit]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]
There are 168 primes less than 1000.[a]
Composite index
[edit]The 128th composite number is 168,[4] one of a few numbers in the list of composites whose indices are the product of strings of digits of in decimal representation.
The first nine with this property are the following:[4]
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
[edit]For the Euler totient there is ,[5] where 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][d]
408,[e] 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]
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,[10] from six numbers (60, 78, 92, 123, 143, and 167).[9]
The difference between 168 and 48 is the factorial of five (120), where their sum is the cube of six (216).
Idoneal number
[edit]Leonhard Euler noted 65 idoneal numbers (the most known, of only a maximum possible of two more), such that for an integer , expressible in only one way, yields a prime power or twice a prime power.[2][11]
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)[12] contains a total of thirty-two divisors whose arithmetic mean is 180[13][14] (the second-largest number to have a totient of 48).[5] Preceding 1848 in the list of idoneal numbers is 1365,[f] whose arithmetic mean of divisors is equal to 168[13][14] (while 1365 has a totient of 576 = 242).
Where 48 is the 27th ideoneal number, 408 is the 58th.[2][g] 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
[edit]In base 10, 168 is the largest of ninety-two known such that does not contain all numerical digits from that base (i.e. 0, 1, 2, ..., 9).[16]
is the first number to have such an expression where between the next two is an interval of ten integers: [70, 79];[16] the median values between these are (75, 74), where the smaller of these two values represents the composite index of 100.[4][h]
Cunningham number
[edit]As a number of the form for positive integers , and not a perfect power, 168 is the thirty-second Cunningham number,[20] where it is one less than a square:
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},[21][22] where 167 represents the thirty-ninth prime[23] (with 39 × 2 = 78). The smallest such chain is {2, 5, 11, 23, 47}.
Eisenstein series
[edit]168 is also coefficient four in the expansion of Eisenstein series ,[24] 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 ;cf.[4] the latter holds a sum-of-divisors of 216 = 63,[9] which is the 168th composite number.[4]
Abstract algebra
[edit]168 is the number of maximal chains in the Bruhat order of symmetric group [25] which is the largest solvable symmetric group with a total of elements.
168 is the order of the second smallest nonabelian simple group 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 ;[26] the Fano plane, isomorphic to the Klein group, has 168 symmetries.
In other fields
[edit]Dominoes
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Dominomatrix.svg/220px-Dominomatrix.svg.png)
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
[edit]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".[27]
Notes
[edit]- ^ (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]
- ^ 32 is the twentieth composite.
- ^ 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] - ^ The latter (210) is the 20th triangle number.[7]
- ^ 505, which is the magic constant of a magic square,[8] is the 408th composite number.
- ^ 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.[15]
- ^ 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]
- ^ 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, ...[17]
which is preceded at 11 with the analogous list of successive super-primes[18] and primes[19] 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
[edit]- ^ Sloane, N. J. A. (ed.). "Sequence A000372 (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.
- ^ a b c d e "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.
- ^ Sloane, N. J. A. (ed.). "Sequence A000396 (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.
- ^ a b c d e f g Sloane, N. J. A. (ed.). "Sequence A002808 (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.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A000010 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A000005 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers: 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
- ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) as n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A000203 (The sum of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
- ^ Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-06.
- ^ 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.
- ^ Sloane, N. J. A. (ed.). "Sequence A005563 (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.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A003601 (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.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A102187 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A006549 (Numbers k such that k and k+1 are prime powers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
- ^ a b "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.
- ^ Sloane, N. J. A. (ed.). "Sequence A059407 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A080262 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A005602 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A348855 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-20.
- ^ Sloane, N. J. A. (ed.). "Sequence A006352 (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.
- ^ Sloane, N. J. A. (ed.). "Sequence A061710 (Number of maximal chains in the Bruhat order of S_n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-01.
- ^ "week214". math.ucr.edu. Retrieved 9 April 2023.
- ^ "$1 Prosperity Forever 168 Note - US Mint". Retrieved 9 April 2023.
External links
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