| Number: | 1728 |
| Divisor: | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728 |
1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross (or grand gross).[1] It is also the number of cubic inches in a cubic foot.
1728 is the cube of 12,[2] and therefore equal to the product of the six divisors of 12 (1, 2, 3, 4, 6, 12).[3] It is also the product of the first four composite numbers (4, 6, 8, and 9), which makes it a compositorial.[4] As a cubic perfect power,[5] it is also a highly powerful number that has a record value (18) between the product of the exponents (3 and 6) in its prime factorization.[6] [7]
It is also a Jordan–Pólya number such that it is a product of factorials:
2! x (3!)2 x 4!=1728
1728 has twenty-eight divisors, which is a perfect count (as with 12, with six divisors). It also has a Euler totient of 576 or 242, which divides 1728 thrice over.[10]
1728 is an abundant and semiperfect number, as it is smaller than the sum of its proper divisors yet equal to the sum of a subset of its proper divisors.[11] [12]
It is a practical number as each smaller number is the sum of distinct divisors of 1728,[13] and an integer-perfect number where its divisors can be partitioned into two disjoint sets with equal sum.[14]
1728 is 3-smooth, since its only distinct prime factors are 2 and 3.[15] This also makes 1728 a regular number[16] which are most useful in the context of powers of 60, the smallest number with twelve divisors:[17]
603=216000=1728 x 125=123 x 53
1728 is also an untouchable number since there is no number whose sum of proper divisors is 1728.[18]
Many relevant calculations involving 1728 are computed in the duodecimal number system, in-which it is represented as "1000".
l{H}:\{\tau\inC,Im(\tau)>0\}
j(\tau)=1728
| |||||||
| \Delta(\tau) |
=1728
| |||||||||||||||
|
Inputting a value of
2i
\tau
i
j(2i)=1728
| |||||||||||||||
|
=663
In moonshine theory, the first few terms in the Fourier q-expansion of the normalized j-invariant exapand as,[20]
1728j(\tau)=1/q+744+196884q+21493760q2+ …
The Griess algebra (which contains the friendly giant as its automorphism group) and all subsequent graded parts of its infinite-dimensional moonshine module hold dimensional representations whose values are the Fourier coefficients in this q-expansion.
The number of directed open knight's tours in
5 x 5
1728 is one less than the first taxicab or Hardy–Ramanujan number 1729, which is the smallest number that can be expressed as sums of two positive cubes in two ways.[22]
Regarding strings of digits of 1728,
1728 is the number of daily chants of the Hare Krishna mantra by a Hare Krishna devotee. The number comes from 16 rounds on a 108 japamala bead.[23]
Equivalently, regular numbers.