1729
This article is about a particular natural number.|View all articles on particular natural numbers
Summary
Names
This number is called the Hardy-Ramanujan number after a conversation between Hardy and Ramanujan where Ramanujan observed that it is the smallest number expressible as the sum of two cubes in two distinct ways: .
Factorization
Properties and families
Property or family | Parameter values | First few members | Proof of membership/containment/satisfaction |
---|---|---|---|
Carmichael number | third among them | 561, 1105, 1729, 2465, 2821, 6601, [SHOW MORE]View list on OEIS | The universal exponent is which divides 1728. |
Poulet number (Fermat pseudoprime to base 2) | sixth among them | 341, 561, 645, 1105, 1387, 1729, 1905, 2047, [SHOW MORE] View list on OEIS | follows from being a Carmichael number. |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
Euler totient function | 1296 | It is the product . |
universal exponent | 36 | It is the least common multiple of . |
divisor count function | 8 | It is the product where the first 1s in each sum represent the multiplicities of the prime divisors. |
divisor sum function | 2240 | It is the product of , , and |
largest prime divisor | 19 | direct from factorization |
largest prime power divisor | 19 | direct from factorization |
square-free part | 1729 | the original number is a square-free number. |
Mobius function | -1 | the number is square-free and has an odd number of prime divisors (namely, 3 prime divisors). |