341: Difference between revisions

Revision as of 21:42, 3 January 2012

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The factorization is as follows, with factors 11 and 31:

$341=11*31$

Property or family	Parameter values	First few members of the family	Proof of satisfaction/membership/containment
Poulet number (also called Sarrus number), i.e., Fermat pseudoprime to base 2	smallest 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	$2^{10}=1024\equiv 1{\pmod {341}}$ , so $2^{340}\equiv 1{\pmod {341}}$ .

Function	Value	Explanation
Euler totient function	300	The Euler totient function is $(11-1)(31-1)=(10)(30)=300$ .
universal exponent	30	The universal exponent is $\operatorname {lcm} \{10,30\}=30$ .
divisor count function	4	$(1+1)(1+1)$ where the first 1s in both factors are the multiplicities of the prime divisors.
divisor sum function	384	$(11^{2}-1)/(11-1)$ times $(31^{2}-1)/(31-1)$ equals $(12)(32)=384$ .
Mobius function	1	The number is square-free and has an even number of prime divisors (2 prime divisors).

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 |-
 | [[satisfies property::Poulet number]] (also called Sarrus number), i.e., [[Fermat pseudoprime]] to base 2 || smallest Poulet number || {{#lst:Poulet number|list}} || <math>2^{10} = 1024 \equiv 1 \pmod {341}</math>, so <math>2^{340} \equiv 1 \pmod {341}</math>.
+|}
+==Arithmetic functions==
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| {{arithmetic function value|Euler totient function|300}} || The Euler totient function is <math>(11 - 1)(31 - 1) = (10)(30) = 300</math>.
+|-
+| {{arithmetic function value|universal exponent|30}} || The universal exponent is <math>\operatorname{lcm}\{10,30 \} = 30</math>.
+|-
+| {{arithmetic function value|divisor count function|4}} || <math>(1 + 1)(1 + 1)</math> where the first 1s in both factors are the multiplicities of the prime divisors.
+|-
+| {{arithmetic function value|divisor sum function|384}} || <math>(11^2-1)/(11-1)</math> times <math>(31^2 - 1)/(31 - 1)</math> equals <math>(12)(32) = 384</math>.
+|-
+| {{arithmetic function value|Mobius function|1}} || The number is square-free and has an even number of prime divisors (2 prime divisors).
 |}