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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
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]
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 for an integer , 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 order42)[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,[f] 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][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).[11]
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).[18]
is the first number to have such an expression where between the next two 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][h]
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,[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 ;cf.[4] the latter holds a sum-of-divisors of 216 = 63,[11] which is the 168th composite number.[4]
^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]
^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, ...[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.
^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. S2CID118287274.
