1728 (number)

1728 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). Great gross (noun) https://www.merriam-webster.com/dictionary/great%20gross 2023-04-04 Merriam-Webster Dictionary It is also the number of cubic inches in a cubic foot. In mathematics 1728 is the cube of 12,A000578 The cubes. 2023-04-03 and therefore equal to the product of the six divisors of 12 (1, 2, 3, 4, 6, 12).A007955 Product of divisors of n. 2023-04-03 It is also the product of the first four composite numbers (4, 6, 8, and 9), which makes it a compositorial.A036691 Compositorial numbers: product of first n composite numbers. 2023-04-03 As a cubic perfect power,A001597 Perfect powers 2023-04-03 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.A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization 2023-04-13 A005361 Product of exponents of prime factorization of n. 2023-04-13 It is also a Jordan–Pólya number such that it is a product of factorials: .A001013 Jordan-Polya numbers: products of factorial numbers 2023-04-03 1728 Numbers Aplenty https://www.numbersaplenty.com/1728 2023-04-04 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.A000010 Euler totient function phi(n): count numbers less than or equal to n and relatively prime to n 2023-04-03 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.A005101 Abundant numbers (sum of divisors of m exceeds 2m). 2023-04-03 A005835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n. 2023-04-03 It is a practical number as each smaller number is the sum of distinct divisors of 1728,A005153 Practical numbers 2023-04-03 and an integer-perfect number where its divisors can be partitioned into two disjoint sets with equal sum.A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum. 2023-04-03 1728 is 3-smooth, since its only distinct prime factors are 2 and 3.A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j greater than or equal to 0. 2023-04-04 This also makes 1728 a regular numberA051037 5-smooth numbers 2023-04-04 Equivalently, regular numbers. which are most useful in the context of powers of 60, the smallest number with twelve divisors:A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. 2023-04-04 . 1728 is also an untouchable number since there is no number whose sum of proper divisors is 1728.A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function. 2023-04-03 Many relevant calculations involving 1728 are computed in the duodecimal number system, in-which it is represented as "1000". 1728 is palindromic in base 11 (133111) and in base 23 (36323). There are 1728 of directed open knight's tours in minichess.A165134 Number of directed Hamiltonian paths in the n X n knight graph 2022-11-30 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 ,10.4153/CMB-1999-050-1 free Bruce C. Berndt Bruce C. Berndt Heng Huat Chan Ramanujan and the modular j-invariant Canadian Mathematical Bulletin 42 4 1999 427–440 1727340 1816362 . Inputting a value of for , where is the imaginary number, yields another cubic integer: . In moonshine theory, the first few terms in the Fourier q-expansion of the normalized j-invariant exapand as,John McKay John McKay (mathematician) Groups and Combinatorics: In memory of Michio Suzuki The Essentials of Monstrous Moonshine Advanced Studies in Pure Mathematics 32 Mathematical Society of Japan Tokyo 2001 351 10.2969/aspm/03210347 978-4-931469-82-2 free 1893502 1015.11012 194379806 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 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.Śrī Dharmavira Prabhu Chanting 64 rounds Harināma daily! Dharmavīra Prahbu Śrī Gaura Radha Govinda International https://www.dharmavira.com/64-rounds-harinam/ live 2023-03-03 en-US 2023-04-04 https://web.archive.org/web/20230404021929/https://www.dharmavira.com/64-rounds-harinam/ See also The year AD 1728 References External links 1728 at Numbers Aplenty. 10 Category:Integers