168 (number) explained

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]

There are 168 primes less than 1000.

Composite index

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:

The next such number is 198 where . The median between twenty-one integers is 58, where 148 is the median of forty-one integers .

Totient and sigma values

there is ,^[5] where is also equivalent to the number of divisors of 168;^[6] only eleven numbers have a totient of 48:.

The number of divisors of 168 is 16,^[7] making it a largely composite number.

408, with a different permutation of the digits where 048 is 48, has an totient of 128. So does the sum-of-divisors of 168,

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,^[8] 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

for an integer , expressible in only one way, yields a prime power or twice a prime power.^[9]

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)^[10] contains a total of thirty-two divisors whose arithmetic mean is 180^{[11] [12]} (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, has a sum-of-divisors of 84 (or, one-half of 168).^[13]

Numbers of the form 2ⁿ

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).^[14] is the first number to have such an expression where between the next two is an interval of ten integers:

Notes and References

1. 2024-04-01 .
2. Web site: 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. 2016-05-28.
3. 2024-04-01 .
4. 2024-04-05 .
5. 2024-04-05 .
6. 2024-04-05 .
7. 2024-04-05 .
8. 2024-07-06 .
9. Euler . Leonard . Illustratio paradoxi circa progressionem numerorum idoneorum sive congruorum . Nova Acta Academiae Scientarum Imperialis Petropolitinae . . 15 . 1806 . 29–32 . math/0507352 . 118287274 .
10. 2024-04-09 .
11. 2023-07-16 .
12. 2023-07-16 .
13. 2024-04-09 .
14. Web site: 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. 2022-12-19.