1728 (number)

← 1727 1728 1729 →
List of numbers Integers ← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinal	one thousand seven hundred twenty-eight
Ordinal	1728th (one thousand seven hundred twenty-eighth)
Factorization	26 × 33
Divisors	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
Greek numeral	,ΑΨΚΗ´
Roman numeral	MDCCXXVIII
Binary	110110000002
Ternary	21010003
Senary	120006
Octal	33008
Duodecimal	100012
Hexadecimal	6C016

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]

See also

• The year AD 1728

References

1. "Great gross (noun)". Merriam-Webster Dictionary . Merriam-Webster, Inc. Retrieved 2023-04-04.
2. Sloane, N. J. A. (ed.). "SequenceA000578(The cubes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-04-03.
3. Sloane, N. J. A. (ed.). "SequenceA007955(Product of divisors of n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-04-03.
4. Sloane, N. J. A. (ed.). "SequenceA036691(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.). "SequenceA001597(Perfect powers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-04-03.
6. Sloane, N. J. A. (ed.). "SequenceA005934(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.). "SequenceA005361(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.). "SequenceA001013(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.). "SequenceA000010(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.). "SequenceA005101(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.). "SequenceA005835(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.). "SequenceA005153(Practical numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-04-03.
14. Sloane, N. J. A. (ed.). "SequenceA083207(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.). "SequenceA003586(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.). "SequenceA051037(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.). "SequenceA000005(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.). "SequenceA005114(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.). "SequenceA165134(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.). "SequenceA011541(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.

External links

• 1728 at Numbers Aplenty.