Search: id:A054269
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%I A054269
%S A054269 1,2,1,4,2,5,1,6,4,5,8,1,3,10,4,5,6,11,10,8,7,4,2,5,11,1,12,6,15,9,12,
%T A054269 6,9,18,9,20,17,18,4,5,14,21,16,13,1,20,26,4,2,5,11,12,17,14,1,12,3,24,
%U A054269 21,13,18,5,14,16,17,11,34,19,14,7,15,4,20,5,30,8,9,21,1,21,18,37,16
%N A054269 Length of period of continued fraction for sqrt(p), p = n-th prime.
%C A054269 The following sequences (allowing offset of first term) all appear to 
               have the same parity: A034953, triangular numbers with prime indices; 
               A054269, length of period of continued fraction for sqrt(p), p prime; 
               A082749, difference between the sum of next prime(n) natural numbers 
               and the sum of next n primes; A006254, numbers n such that 2n-1 is 
               prime; A067076, 2n+3 is a prime. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), 
               Sep 10 2004
%C A054269 Note that primes of the form n^2+1 (A002496) have a continued fraction 
               whose period length is 1; odd primes of the form n^2+2 (A056899) 
               have length 2; odd primes of the form n^2-2 (A028871) have length 
               4. - T. D. Noe, Nov 03 2006
%C A054269 For an odd prime p, the length of the period is odd if p=1 (mod 4) or 
               even if p=3 (mod 4). - T. D. Noe (noe(AT)sspectra.com), May 22 2007
%H A054269 T. D. Noe, <a href="b054269.txt">Table of n, a(n) for n=1..10000</a>
%H A054269 A. I. Gliga, <a href="http://www.math.princeton.edu/mathlab/jr02fall/
               Periodicity/alexajp.pdf">On continued fractions of the square root 
               of prime numbers</a>
%p A054269 with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 
               'periodic','quotients'); printf(`%d,`, nops(cfr[2])) od:
%t A054269 Table[p=Prime[n]; Length[Last[ContinuedFraction[Sqrt[p]]]],{n,100}] - 
               T. D. Noe (noe(AT)sspectra.com), May 22 2007
%Y A054269 Cf. A003285.
%Y A054269 Cf. A130272 (primes at which the period length sets a new record).
%Y A054269 Sequence in context: A105474 A120988 A095979 this_sequence A086450 A106044 
               A124896
%Y A054269 Adjacent sequences: A054266 A054267 A054268 this_sequence A054270 A054271 
               A054272
%K A054269 nonn,easy,nice
%O A054269 1,2
%A A054269 N. J. A. Sloane (njas(AT)research.att.com), May 05 2000
%E A054269 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 05 2000

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