A010129 -id:A010129 - cp's OEIS frontend

A010486 Decimal expansion of square root of 31.

5, 5, 6, 7, 7, 6, 4, 3, 6, 2, 8, 3, 0, 0, 2, 1, 9, 2, 2, 1, 1, 9, 4, 7, 1, 2, 9, 8, 9, 1, 8, 5, 4, 9, 5, 2, 0, 4, 7, 6, 3, 9, 3, 3, 7, 7, 5, 7, 0, 4, 1, 4, 3, 0, 3, 9, 6, 8, 4, 3, 2, 5, 8, 5, 6, 0, 3, 5, 8, 9, 8, 3, 9, 2, 5, 4, 2, 3, 6, 2, 9, 2, 9, 2, 7, 2, 1, 8, 3, 9, 6, 1, 8, 4, 9, 2, 6, 6, 7

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {1, 1, 3, 5, 3, 1, 1, 10} repeated. - Harry J. Smith, Jun 04 2009

			5.567764362830021922119471298918549520476393377570414303968432585603589....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A010129 (continued fraction). - Harry J. Smith, Jun 04 2009

RealDigits[N[Sqrt[31], 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(31); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010486.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

A041051 Denominators of continued fraction convergents to sqrt(31).

Original entry on oeis.org

1, 1, 2, 7, 37, 118, 155, 273, 2885, 3158, 6043, 21287, 112478, 358721, 471199, 829920, 8770399, 9600319, 18370718, 64712473, 341933083, 1090511722, 1432444805, 2522956527, 26662010075, 29184966602, 55846976677, 196725896633, 1039476459842, 3315155276159

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,3040,0,0,0,0,0,0,0,-1).

Cf. A041050, A010129, A020788, A010486.

I:=[1, 1, 2, 7, 37, 118, 155, 273, 2885, 3158, 6043, 21287, 112478, 358721, 471199, 829920]; [n le 16 select I[n] else 3040*Self(n-8) - Self(n-16): n in [1..50]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[31],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
Denominator[Convergents[Sqrt[31], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

G.f.: -(x^14 -x^13 +2*x^12 -7*x^11 +37*x^10 -118*x^9 +155*x^8 -273*x^7 -155*x^6 -118*x^5 -37*x^4 -7*x^3 -2*x^2 -x -1) / (x^16 -3040*x^8 +1). - Colin Barker, Nov 12 2013

More terms from Colin Barker, Nov 12 2013

A307453 a(n) is the least prime p for which the continued fraction expansion of sqrt(p) has exactly n consecutive 1's starting at position 2.

Original entry on oeis.org

2, 3, 31, 7, 13, 3797, 5273, 4987, 90371, 79873, 2081, 111301, 1258027, 5325101, 12564317, 9477889, 47370431, 709669249, 1529640443, 2196104969, 392143681, 8216809361, 30739072339, 200758317433, 370949963971, 161356959383, 1788677860531, 7049166342469, 4484287435283, 3690992602753

Offset: 0

Michel Marcus, Apr 09 2019

nonn

			For p = 2,  we have [1; 2, ...]; see A040000.
For p = 3,  we have [1; 1, 2, ...]; see A040001.
For p = 31, we have [5; 1, 1, 3, ...]; see A010129.
For p = 7,  we have [2; 1, 1, 1, 4, ...]; see A010121.

Piotr Miska, Maciej Ulas, On consecutive 1's in continued fractions expansions of square roots of prime numbers, arXiv:1904.03404 [math.NT], 2019. See Table 1 p. 15.
Index entries for continued fractions for constants

Cf. A040000, A040001, A010121, A010129.

isok(p, n) = {my(c=contfrac(sqrt(p)));  for (k=2, n+1, if (c[k] != 1, return (0));); return(c[n+2] !=  1);}
a(n) = {my(p=2); while (! isok(p, n), p = nextprime(p+1)); p;}

Limit_{n->infinity} (sqrt(a(n)) - floor(sqrt(a(n)))) = A094214. - Daniel Suteu, Apr 09 2019

a(21)-a(29) from Daniel Suteu, Apr 09 2019

a(0) added by Chai Wah Wu, Apr 09 2019

cp's OEIS Frontend

A010486 Decimal expansion of square root of 31.

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A041051 Denominators of continued fraction convergents to sqrt(31).

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A307453 a(n) is the least prime p for which the continued fraction expansion of sqrt(p) has exactly n consecutive 1's starting at position 2.

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