← 1727 1728 1729 →
Cardinalone thousand seven hundred twenty-eight
Ordinal1728th
(one thousand seven hundred twenty-eighth)
Factorization26 × 33
Divisors1, 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
Greek numeral,ΑΨΚΗ´
Roman numeralMDCCXXVIII
Binary110110000002
Ternary21010003
Senary120006
Octal33008
Duodecimal100012
Hexadecimal6C016

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.

## In mathematics

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]

{\displaystyle {\begin{aligned}1728&=3^{3}\times 4^{3}=2^{3}\times 6^{3}={\mathbf {12^{3))}\\1728&=6^{3}+8^{3}+10^{3}\\1728&=24^{2}+24^{2}+24^{2}\\\end{aligned))}

It is also a Jordan–Pólya number such that it is a product of factorials: ${\displaystyle 2!\times (3!)^{2}\times 4!=1728.}$[8][9]

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]

${\displaystyle 60^{3}=216000=1728\times 125=12^{3}\times 5^{3))$

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".

### Modular j-invariant

1728 occurs in the algebraic formula for the j-invariant of an elliptic curve, as a function over a complex variable on the upper half-plane ${\displaystyle \,{\mathcal {H)):\{\tau \in \mathbb {C} ,{\text{ ))\mathrm {Im} (\tau )>0\))$,[19]

${\displaystyle j(\tau )=1728{\frac {g_{2}(\tau )^{3)){\Delta (\tau )))=1728{\frac {g_{2}(\tau )^{3)){g_{2}(\tau )^{3}-27g_{3}(\tau )^{2))}.}$

Inputting a value of ${\displaystyle 2i}$ for ${\displaystyle \tau }$, where ${\displaystyle i}$ is the imaginary number, yields another cubic integer:

${\displaystyle j(2i)=1728{\frac {g_{2}(2i)^{3)){g_{2}(2i)^{3}-27g_{3}(2i)^{2))}=66^{3}.}$

In moonshine theory, the first few terms in the Fourier q-expansion of the normalized j-invariant exapand as,[20]

${\displaystyle 1728{\text{ ))j(\tau )=1/q+744+196884q+21493760q^{2}+\cdots }$

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.

### Other properties

The number of directed open knight's tours in ${\displaystyle 5\times 5}$ minichess is 1728.[21]

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]

## In culture

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]

## References

1. ^ "Great gross (noun)". Merriam-Webster Dictionary. Merriam-Webster, Inc. Retrieved 2023-04-04.
2. ^ Sloane, N. J. A. (ed.). "Sequence A000578 (The cubes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
3. ^ Sloane, N. J. A. (ed.). "Sequence A007955 (Product of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
4. ^ Sloane, N. J. A. (ed.). "Sequence A036691 (Compositorial numbers: product of first n composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
5. ^ Sloane, N. J. A. (ed.). "Sequence A001597 (Perfect powers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
6. ^ Sloane, N. J. A. (ed.). "Sequence A005934 (Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-13.
7. ^ Sloane, N. J. A. (ed.). "Sequence A005361 (Product of exponents of prime factorization of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-13.
8. ^ Sloane, N. J. A. (ed.). "Sequence A001013 (Jordan-Polya numbers: products of factorial numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
9. ^ "1728". Numbers Aplenty. Retrieved 2023-04-04.
10. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and relatively prime to n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
11. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
12. ^ Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
13. ^ Sloane, N. J. A. (ed.). "Sequence A005153 (Practical numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
14. ^ Sloane, N. J. A. (ed.). "Sequence A083207 (Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
15. ^ Sloane, N. J. A. (ed.). "Sequence A003586 (3-smooth numbers: numbers of the form 2^i*3^j with i, j greater than or equal to 0.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
16. ^ Sloane, N. J. A. (ed.). "Sequence A051037 (5-smooth numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
Equivalently, regular numbers.
17. ^ 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 2023-04-04.
18. ^ Sloane, N. J. A. (ed.). "Sequence A005114 (Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.
19. ^ Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1. MR 1727340. S2CID 1816362.
20. ^ John McKay (2001). "The Essentials of Monstrous Moonshine". Groups and Combinatorics: In memory of Michio Suzuki. Advanced Studies in Pure Mathematics. Vol. 32. Tokyo: Mathematical Society of Japan. p. 351. doi:10.2969/aspm/03210347. ISBN 978-4-931469-82-2. MR 1893502. S2CID 194379806. Zbl 1015.11012.
21. ^ Sloane, N. J. A. (ed.). "Sequence A165134 (Number of directed Hamiltonian paths in the n X n knight graph)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-30.
22. ^ Sloane, N. J. A. (ed.). "Sequence A011541 (Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-30.
23. ^ Śrī Dharmavira Prabhu. "Chanting 64 rounds Harināma daily!". Dharmavīra Prahbu. Śrī Gaura Radha Govinda International. Archived from the original on 2023-04-04. Retrieved 2023-03-03.