1728 (number)

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← 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, mdccxxviii
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]

1728 = 3 3 × 4 3 = 2 3 × 6 3 = 12 3 1728 = 6 3 + 8 3 + 10 3 1728 = 24 2 + 24 2 + 24 2 {\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}}} {\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: 2 ! × ( 3 ! ) 2 × 4 ! = 1728 {\displaystyle 2!\times (3!)^{2}\times 4!=1728} {\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 an 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]

60 3 = 216000 = 1728 × 125 = 12 3 × 5 3 {\displaystyle 60^{3}=216000=1728\times 125=12^{3}\times 5^{3}} {\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 H : { τ ∈ C ,   I m ( τ ) > 0 } {\displaystyle \,{\mathcal {H}}:\{\tau \in \mathbb {C} ,{\text{ }}\mathrm {Im} (\tau )>0\}} {\displaystyle \,{\mathcal {H}}:\{\tau \in \mathbb {C} ,{\text{ }}\mathrm {Im} (\tau )>0\}},[19]

j ( τ ) = 1728 g 2 ( τ ) 3 Δ ( τ ) = 1728 g 2 ( τ ) 3 g 2 ( τ ) 3 − 27 g 3 ( τ ) 2 {\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}}}} {\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 2 i {\displaystyle 2i} {\displaystyle 2i} for τ {\displaystyle \tau } {\displaystyle \tau }, where i {\displaystyle i} {\displaystyle i} is the imaginary number, yields another cubic integer:

j ( 2 i ) = 1728 g 2 ( 2 i ) 3 g 2 ( 2 i ) 3 − 27 g 3 ( 2 i ) 2 = 66 3 {\displaystyle j(2i)=1728{\frac {g_{2}(2i)^{3}}{g_{2}(2i)^{3}-27g_{3}(2i)^{2}}}=66^{3}} {\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]

1728   j ( τ ) = 1 / q + 744 + 196884 q + 21493760 q 2 + ⋯ {\displaystyle 1728{\text{ }}j(\tau )=1/q+744+196884q+21493760q^{2}+\cdots } {\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 5 × 5 {\displaystyle 5\times 5} {\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]

Decimal digits

Regarding strings of digits of 1728,

  • The sum between 1 and 7 inclusive (as a triangular number) yields 28.
  • Where 1728 is the cube of 12, the sum 1 + 728 = 729 = 93. The digit sum of 1728 is 18.
  • The product of the digits of 1728 is 112, as with 744.

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-bead japamala.[23]

See also

References

  1. "Great gross (noun)". Merriam-Webster Dictionary. 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.