561: Difference between revisions

Revision as of 21:10, 2 January 2012

This article is about a particular natural number.|View all articles on particular natural numbers

The factorization into primes is:

$\!561=3*11*17$

Property or family	Parameter values	First few numbers satisfying the property
Carmichael number (also called absolute pseudoprime)	first Carmichael number	561, 1105, 1729
Poulet number (Fermat pseudoprime to base 2)	second Poulet number	341, 561, 645

Function	Value	Explanation
Euler totient function	320	The Euler totient function is $(3-1)(11-1)(17-1)=(2)(10)(16)=320$ .
universal exponent	80	The universal exponent is the least common multiple of $\{3-1,11-1,17-1\}$ , which is $\operatorname {lcm} \{2,10,16\}$ and equals 80. Note that 561 is a Carmichael number precisely because the universal exponent divides 561 - 1 = 560.
Mobius function	-1	The number is a square-free number and it has an odd number of prime divisors (3 prime divisors).
divisor count function	8	$(1+1)(1+1)(1+1)$ where the first 1s in each sum denote the multiplicities of the prime divisors.
largest prime divisor	17
largest prime power divisor	17

@@ Line 16: / Line 16: @@
 | [[satisfies property::Carmichael number]] (also called absolute pseudoprime) || first Carmichael number || [[561]], [[1105]], [[1729]]
 |-
-| [[satisfies property::Poulet number]] ([[Fermat pseudoprime to base 2) || second Poulet number || [[341]], [[561]], [[645]]
+| [[satisfies property::Poulet number]] ([[Fermat pseudoprime]] to base 2) || second Poulet number || [[341]], [[561]], [[645]]
+|}
+==Arithmetic functions==
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| {{arithmetic function value|Euler totient function|320}} || The Euler totient function is <math>(3 - 1)(11 - 1)(17 - 1) = (2)(10)(16) = 320</math>.
+|-
+| {{arithmetic function value|universal exponent|80}} || The universal exponent is the [[least common multiple]] of <math>\{3 - 1, 11 - 1, 17 - 1\}</math>, which is <math>\operatorname{lcm} \{ 2,10, 16\}</math> and equals 80.<br>Note that 561 is a [[Carmichael number]] precisely because the universal exponent divides 561 - 1 = 560.
+|-
+| {{arithmetic function value|Mobius function|-1}} || The number is a [[square-free number]] and it has an odd number of prime divisors (3 prime divisors).
+|-
+| {{arithmetic function value|divisor count function|8}} || <math>(1 + 1)(1 + 1)(1 + 1)</math> where the first 1s in each sum denote the multiplicities of the prime divisors.
+|-
+| {{arithmetic function value|largest prime divisor|17}} ||
+|-
+| {{arithmetic function value|largest prime power divisor|17}} ||
 |}