# 1729

## 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).