1729: Difference between revisions

Latest revision as of 20:18, 14 July 2024

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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 members	Proof of membership/containment/satisfaction
Carmichael number	third among them	561, 1105, 1729, 2465, 2821, 6601, [SHOW MORE] 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461 View list on OEIS	The universal exponent is $\operatorname {lcm} \{6,12,18\}=36$ which divides 1728.
Poulet number (Fermat pseudoprime to base 2)	sixth among them	341, 561, 645, 1105, 1387, 1729, 1905, 2047, [SHOW MORE] 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341 View list on OEIS	follows from being a Carmichael number.

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).

@@ Line 5: / Line 5: @@
 ===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: <math>\! 1729 = 10^3 + 9^23 = 12^3 + 1^3</math>.
+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: <math>\! 1729 = 10^3 + 9^3 = 12^3 + 1^3</math>.
 ===Factorization===
@@ Line 14: / Line 14: @@
 {| class="sortable" border="1"
-! Property or family !! Parameter values !! First few numbers
+! Property or family !! Parameter values !! First few members !! Proof of membership/containment/satisfaction
 |-
-| [[Carmichael number]] || third among them || [[561]], [[1105]], [[1729]], ...
+| [[satsfies property::Carmichael number]] || third among them || {{#lst:Carmichael number|list}} || The universal exponent is <math>\operatorname{lcm}\{ 6, 12, 18\} = 36</math> which divides 1728.
 |-
-| [[Poulet number]] ([[Fermat pseudoprime to base 2) || sixth among them || [[341]], [[561]], [[645]], [[1105]], [[1387]], [[1729]], [[1905]], [[2047]]
+| [[satisfies property::Poulet number]] ([[Fermat pseudoprime]] to base 2) || sixth among them || {{#lst:Poulet number|list}} || follows from being a Carmichael number.
+|}
+==Arithmetic functions==
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| {{arithmetic function value|Euler totient function|1296}} || It is the product <math>(7 - 1)(13 - 1)(19 - 1) = (6)(12)(18) = 1296</math>.
+|-
+| {{arithmetic function value|universal exponent|36}} || It is the [[least common multiple]] of <math>\{7 - 1, 13  - 1, 19 - 1\}</math>.
+|-
+| {{arithmetic function value|divisor count function|8}} || It is the product <math>(1 + 1)(1 + 1)(1 + 1)</math> where the first 1s in each sum represent the multiplicities of the prime divisors.
+|-
+| {{arithmetic function value|divisor sum function|2240}} || It is the product of <math>(7^2 - 1)/(7 - 1)</math>, <math>(13^2 - 1)/(13 - 1)</math>, and <math>(19^2 - 1)/(19 - 1)</math>
+|-
+| {{arithmetic function value|largest prime divisor|19}} || direct from factorization
+|-
+| {{arithmetic function value|largest prime power divisor|19}} || direct from factorization
+|-
+| {{arithmetic function value|square-free part|1729}} || the original number is a [[square-free number]].
+|-
+| {{arithmetic function value|Mobius function|-1}} || the number is square-free and has an odd number of prime divisors (namely, 3 prime divisors).
 |}