168 (number)

← 167 168 169 →
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

.

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.^[a]

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]

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

${\displaystyle \varphi (n)}$ 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][d]

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

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

Idoneal number

(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

= 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

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][h]

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 = 6³,^[9] which is the 168th composite number.^[4]

Abstract algebra

168 is the number of

${\displaystyle \mathrm {S} _{4},}$[24] which is the largest solvable symmetric group with a total of ${\displaystyle 4!=24}$ elements.

168 is the order of the

${\displaystyle \mathrm {PSL} (2,7).}$ From , whose symmetry group is ${\displaystyle \mathrm {PSL} (2,7)}$;

symmetries

.

In other fields

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

Notes

References

1. ^ 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.
2. ^ ^{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.
3. ^ 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.
4. ^ ^{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.
5. ^ ^{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.
6. ^ 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.
7. ^ 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.
8. ^ 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.
9. ^ ^{a b c} 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.
10. S2CID 118287274
    .
11. ^ 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.
12. ^ ^{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.
13. ^ ^{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.
14. ^ 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.
15. ^ ^{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.
16. ^ 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.
17. ^ 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.
18. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
19. ^ 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.
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.
21. ^ 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.
22. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-20.
23. ^ 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.
24. ^ 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.
25. ^ "week214". math.ucr.edu. Retrieved 9 April 2023.
26. ^ "$1 Prosperity Forever 168 Note - US Mint". Retrieved 9 April 2023.

External links

Wikimedia Commons has media related to 168 (number).

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