A130272 -id:A130272 - cp's OEIS frontend

A054269 Length of period of continued fraction for sqrt(prime(n)).

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

Offset: 1

N. J. A. Sloane, May 05 2000

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, 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, May 22 2007

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

Cf. A003285, A130272 (primes at which the period length sets a new record).

with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 'periodic','quotients'); printf(`%d,`, nops(cfr[2])) od:

Table[p=Prime[n]; Length[Last[ContinuedFraction[Sqrt[p]]]],{n,100}] (* T. D. Noe, May 22 2007 *)
Length[ContinuedFraction[Sqrt[#]][[2]]]&/@Prime[Range[100]] (* Harvey P. Dale, Sep 28 2024 *)

More terms from James Sellers, May 05 2000

A059800 Smallest prime p such that the quotient-cycle length in continued fraction expansion of sqrt(p) is n: smallest prime p(m) for which A054269(m)=n.

Original entry on oeis.org

2, 3, 41, 7, 13, 19, 73, 31, 113, 43, 61, 103, 193, 179, 109, 191, 157, 139, 337, 151, 181, 491, 853, 271, 457, 211, 1109, 487, 821, 379, 601, 463, 613, 331, 1061, 1439, 421, 619, 541, 1399, 1117, 571, 1153, 823, 1249, 739, 1069, 631, 1021, 1051, 1201

Offset: 1

Labos Elemer, Feb 23 2001

nonn

			The quotient-cycle length L=9=A054269(m) first appears for p(30)=113, so a(9)=113 namely, at first A054269(30)=9; a(A054269(30)) = p(30) = 113 = a(9). The quotient cycle with L=16 first emerges for sqrt(191) and it is: cfrac(sqrt(191), 'periodic', 'quotients')= [[13],[1,4,1,1,3,2,2,13,2 2,3,1,1,4,1,26]].

T. D. Noe, Table of n, a(n) for n = 1..2000

Cf. A054269.

Cf. A013646, A130272

a(n) = Min{p|A054269(sequence number of p)=n; p is prime}.

cp's OEIS Frontend

A054269 Length of period of continued fraction for sqrt(prime(n)).

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