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: $\!1729=10^{3}+9^{3}=12^{3}+1^{3}$.

Factorization

$\!1729=7\cdot 13\cdot 19$

Properties and families

Property or family	Parameter values	First few numbers
Carmichael number	third among them	561, 1105, 1729, ...
Poulet number (Fermat pseudoprime to base 2)	sixth among them	341, 561, 645, 1105, 1387, 1729, 1905, 2047

Arithmetic functions

Function	Value	Explanation
Euler totient function	1296	It is the product $(7-1)(13-1)(19-1)=(6)(12)(18)=1296$.
universal exponent	36	It is the least common multiple of $\{7-1,13-1,19-1\}$.
divisor count function	8	It is the product $(1+1)(1+1)(1+1)$ where the first 1s in each sum represent the multiplicities of the prime divisors.
divisor sum function	2240	It is the product of $(7^{2}-1)/(7-1)$, $(13^{2}-1)/(13-1)$, and $(19^{2}-1)/(19-1)$
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).