A040001 -id:A040001 - cp's OEIS frontend

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.

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

A084642 A Jacobsthal ratio.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 0

Paul Barry, Jun 08 2003

The Jacobsthal recurrence means that A001045(n+1)/A001045(n) = 1 + 2/(A001045(n)/A001045(n-1)). The sequence of these fractions alternates after the first terms values just above 2 and just below 2, because the mapping x -> 1+2/x is concave in the neighborhood of x=2, where x=2 is an attractor. As a consequence, this sequence here iterates like A040001 or A000034 after a few terms. - R. J. Mathar, Sep 17 2008

Decimal expansion of 433/3300. - Elmo R. Oliveira, May 06 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A001045, A040001.

[1,3] cat [1+ (n mod 2): n in [2..120]]; // G. C. Greubel, Mar 20 2023

Table[(3-(-1)^n)/2 +Boole[n==1], {n,0,120}] (* G. C. Greubel, Mar 20 2023 *)

[1 + (n%2) + int(n==1) for n in range(121)] # G. C. Greubel, Mar 20 2023

a(n) = floor(A001045(n+2)/A001045(n+1)).
a(n) = floor((2^(n+2) - (-1)^(n+2))/(2^(n+1) - (-1)^(n+1))).
From G. C. Greubel, Mar 20 2023: (Start)
a(n) = A000034(n) + [n=1].
a(n) = a(n-2), for n > 3, with a(0) = 1, a(1) = 3, a(2) = 1, a(3) = 2.
G.f.: (1 + 3*x - x^3)/(1-x^2).
E.g.f.: (1/2)*(2*x + 3*exp(x) - exp(-x)). (End)

A154388 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Philippe Deléham, Jan 08 2009

			Triangle begins:
  1;
  0, 1;
  0, 1, 0;
  0, 0, 0, 1;
  0, 0, 0, 1, 0;
  0, 0, 0, 0, 0, 1; ...

Sum_{k=0..n} T(n,k)*x^(n-k) = A135528(n+1), A000012(n), A040001(n), A153284(n+1) for x = 0,1,2,3 respectively.
G.f.: (1+y*x+(y-y^2)*x^2)/(1-y^2*x^2). - Philippe Deléham, Dec 17 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000012(n), A158302(n) for x = 0, 1, 2 respectively. - Philippe Deléham, Dec 17 2011

A175921 Period 5: repeat [1, 2, 2, -1, 1].

Original entry on oeis.org

1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1

Offset: 1

Jaroslav Krizek, Oct 17 2010

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A040001, A028356, A142150, A000012, A175922.

PadRight[{},120,{1,2,2,-1,1}] (* Harvey P. Dale, Dec 30 2016 *)

a(n)=[1,1,2,2,-1][n%5+1] \\ Charles R Greathouse IV, Mar 12 2017

Edited by Joerg Arndt, Sep 16 2013

A064849 Period of continued fraction for sqrt(3)*n.

Original entry on oeis.org

2, 2, 2, 2, 4, 8, 2, 4, 10, 4, 2, 8, 6, 2, 2, 8, 6, 20, 4, 8, 14, 8, 8, 12, 16, 2, 30, 2, 16, 4, 18, 16, 10, 20, 16, 20, 18, 8, 6, 12, 2, 12, 8, 8, 6, 20, 20, 20, 30, 36, 6, 2, 8, 68, 14, 2, 16, 32, 22, 4, 38, 18, 40, 36, 6, 28, 10, 20, 40, 8, 4, 40, 18, 22, 16, 12, 28, 2, 46, 20, 98, 8

Offset: 1

R. K. Guy, Oct 26 2001

nonn

			A040001 (cfrac for n=1) has period length 2, so a(1)=2. A040008 (cfrac for n=2) has period length 2, so a(2)=2. A040021 (cfrac for =3) has period length 2, so a(3)=2. - _R. J. Mathar_, Feb 10 2016

Table[Length[Last[ContinuedFraction[Sqrt[3] n]]], {n, 128}]

A307530 Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.

Original entry on oeis.org

3, 23, 47, 59, 61, 79, 97, 137, 139, 163, 167, 191, 193, 223, 251, 281, 283, 313, 317, 349, 353, 359, 389, 397, 431, 433, 439, 479, 521, 523, 563, 569, 571, 613, 617, 619, 659, 661, 673, 719, 727, 769, 773, 823, 827, 829, 839, 881, 883, 887, 941, 947, 953, 1009

Offset: 1

Michel Marcus, Apr 13 2019

nonn

Misak and Ulas prove that the density of primes with k ones is 1/(Fibonacci(k+3)*Fibonacci(k+1)) = 1/3, here with k=1 (a single 1).

			For p = 3,  we have [1; 1, 2, ...]; see A040001.
For p = 27, we have [4; 1, 3, ...]; see A010127.
For p = 47, we have [6; 1, 5, ...]; see A010137.

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 Corollary 4.3. p. 13.
Index entries for continued fractions for constants

Cf. A040001, A010127, A010137, A307453.

isok(p) = my(cf = contfrac(sqrt(p))); (cf[2] == 1) && (cf[3] != 1);
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", ")));

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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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A084642 A Jacobsthal ratio.

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A154388 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A175921 Period 5: repeat [1, 2, 2, -1, 1].

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A064849 Period of continued fraction for sqrt(3)*n.

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Mathematica

A307530 Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.

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PARI