1105: Difference between revisions

Latest revision as of 21:17, 3 January 2012

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Summary

Factorization

1105 is a square-free number. It has prime factors 5, 13, and 17:

$\!1105=5*13*17$

Properties and families

Property or family	Parameter values	First few numbers satisfying the property	Proof of satisfaction/containment/membership
Carmichael number (also called absolute pseudoprime)	second Carmichael number	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 $\operatorname {lcm} \{5-1,13-1,17-1\}=48$ which divides 1105 - 1 = 1104.
Poulet number (also called Sarrus number) (Fermat pseudoprime to base 2)	fourth Poulet number	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 its being a Carmichael number (note that Carmichael number is odd).

Arithmetic functions

Function	Value	Explanation
Euler totient function	768	The Euler totient function is $(5-1)(13-1)(17-1)=(4)(12)(16)=768$ .
universal exponent	48	The universal exponent is $\operatorname {lcm} \{5-1,13-1,17-1\}=\operatorname {lcm} \{4,12,16\}=48$ .
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 12: / Line 12: @@
 {| class="sortable" border="1"
-! Property or family !! Parameter values !! First few numbers satisfying the property
+! Property or family !! Parameter values !! First few numbers satisfying the property !! Proof of satisfaction/containment/membership
 |-
-| [[satisfies property::Carmichael number]] (also called absolute pseudoprime) || second Carmichael number || {{#lst:Carmichael number|list}}
+| [[satisfies property::Carmichael number]] (also called absolute pseudoprime) || second Carmichael number || {{#lst:Carmichael number|list}} || The [[universal exponent]] is <math>\operatorname{lcm} \{ 5-1,13-1,17-1 \} = 48</math> which divides 1105 - 1 = 1104.
 |-
-| [[satisfies property::Poulet number]] (also called Sarrus number) ([[Fermat pseudoprime]] to base 2) || fourth Poulet number || {{#lst:Poulet number|list}}
+| [[satisfies property::Poulet number]] (also called Sarrus number) ([[Fermat pseudoprime]] to base 2) || fourth Poulet number || {{#lst:Poulet number|list}} || follows from its being a Carmichael number (note that [[Carmichael number is odd]]).
+|}
+==Arithmetic functions==
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| {{arithmetic function value|Euler totient function|768}} || The Euler totient function is <math>(5-1)(13-1)(17-1) = (4)(12)(16) = 768</math>.
+|-
+| {{arithmetic function value|universal exponent|48}} || The universal exponent is <math>\operatorname{lcm}\{5-1,13-1,17-1\} = \operatorname{lcm}\{4,12,16\} = 48</math>.
+|-
+| {{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}} ||
 |}