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Search: id:A054269
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| A054269 |
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Length of period of continued fraction for sqrt(p), p = n-th prime. |
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+0
15
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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
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
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
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
A. I. Gliga, On continued fractions of the square root of prime numbers
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MAPLE
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with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 'periodic', 'quotients'); printf(`%d, `, nops(cfr[2])) od:
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MATHEMATICA
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Table[p=Prime[n]; Length[Last[ContinuedFraction[Sqrt[p]]]], {n, 100}] - T. D. Noe (noe(AT)sspectra.com), May 22 2007
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CROSSREFS
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Cf. A003285.
Cf. A130272 (primes at which the period length sets a new record).
Sequence in context: A105474 A120988 A095979 this_sequence A086450 A106044 A124896
Adjacent sequences: A054266 A054267 A054268 this_sequence A054270 A054271 A054272
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), May 05 2000
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 05 2000
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