login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A334116 a(n) is the least number k greater than n such that the square roots of both k and n have continuous fractions with the same period p and, if p > 1, the same periodic terms except for the last term. 1
1, 5, 8, 4, 10, 12, 32, 15, 9, 17, 40, 20, 74, 33, 24, 16, 26, 39, 1880, 30, 112, 660, 96, 35, 25, 37, 104, 299, 338, 42, 77600, 75, 60, 78, 48, 36, 50, 84, 68, 87, 130, 56, 288968, 468, 350, 3242817, 192, 63, 49, 65, 200, 2726, 1042, 1628, 180, 72, 308, 425, 5880, 95 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Note that a(n)=n if n is a square. The square root of a squarefree integer n has a continued fraction of the form [e(0);[e(1),...,e(p)]] with e(p)=2e(0) and e(i)=e(p-i) for 0 < i < p, see reference. The symmetric part [e(1),...,e(p-1)] of the continued fraction [m;[e(1),...,e(p-1), 2m]] will be called the pattern of n. 2 has the empty pattern (sqrt(2)=[1,[2]]), 3 has the pattern [1] (sqrt(3)=[1,[1,2]]) and so on. In this sense, the description of the sequence can be simplified as "Least number greater than n with the same pattern".

It can be can proved (see link) that integers with the same pattern are terms of a quadratic sequence.

An ambiguity has to be fixed: sqrt(2)=[1,[2]] = [1,[2,2]] = [1,[2,2,2]] and so on. We define that the shortest pattern is correct, here it is empty. Comment on the third subsequence (2),6,12,... below: The second term 6 has the pattern [2], but the first term 2 in brackets has the "wrong" pattern, after fixing the ambiguity.

REFERENCES

Kenneth H. Rosen, Elementary number theory and its applications, Addison-Wesley, 3rd ed. 1993, page 428.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..138

Gerhard Kirchner, Continued fractions with the same pattern

EXAMPLE

1) p=1: f(1)=2, f(2)=a(2)=5, f(3)=a(5)=10, f(4)=a(10)=17,..

sqrt(2)=[1,[2]], sqrt(5)=[2,[4]], sqrt(10)=[3,[6]], sqrt(17)=[4,[8]],..

2) p=2: f(1)=3, f(2)=a(3)=8, f(3)=a(8)=15, f(4)=a(15)=24,..

sqrt(3)=[1,[1,2]], sqrt(8)=[2,[1,4]], sqrt(15)=[3,[1,6]], sqrt(24)=[4,[1,8]],..

3) p=3: f(1)=41, f(2)=a(41)=130, f(3)=a(130)=269,..

sqrt(41)=[6,[2,2,12]], sqrt(130)=[11,[2,2,121]], sqrt(269)=[16,[2,2,256]],..

4) p=4: f(1)=33, f(2)=a(33)=60, f(3)=a(60)=95,..

sqrt(33)=[5,[1,2,1,10]], sqrt(60)=[7,[1,2,1,49]], sqrt(95)=[9,[1,2,1,81]],..

Several subsequences f(k) with f(k+1)=a(f(k)).

k>1 if first term in brackets, k>0 otherwise.

First terms  Period  Formula           Example

1) 2,5,10,17   1  A002522(k)=k^2+1           1

2) 3,8,15,24   2  A005563(k)=(k+1)^2-1       2

3)(2),6,12     2  A002378(k)=k*(k+1)

4) 7,32,75     4  A013656(k)=k*(9*k-2)

5) 11,40,87    2  A147296(k)=k*(9*k+2)

6) 13,74,185   5  A154357(k)=25*k^2-14*k+2

7) (3),14,33   4  A033991(k)=k*(4*k-1)       4

8) (5),18,39   2  A007742(k)=k*(4*k+1)

9) 21,112,275  6  A157265(k)=36*k^2-17*k+2

10)23,96,219   4  A154376(k)=25*k^2-2*k

11)27,104,231  2  A154377(k)=25*k^2+2*k

12)28,299,858  4  A156711(k)=144*k^2-161*k+45

13)29,338,985  5  A156640(k)=169*k^2+140*k+29

14)(8),34,78   4  A154516(k)=9*k^2-k

15)(10),38,84  2  A154517(k)=9*k^2+k

16)(2),41,130  3  A154355(k)=25*k^2-36*k+13  3

17)47,192,435  4  A157362(k)=49*k^2-2*k

PROG

(Maxima)

block([nmax: 100],

/*saves the first nmax terms in the current directory*/

algebraic: true, local(coeff), showtime: true,

fl: openw(sconcat("terms", nmax, ".txt")),

coeff(w, m):=

  block(a: m, p: 0, s: w, vv:[],

   while a<2*m do

    (p: p+1, s: ratsimp(1/(s-floor(s))), a: floor(s),

     if a<2*m then vv: append(vv, [a])),

   j: floor((p-1)/2),

   if mod(p, 2)=0 then v: [1, 0, vv[j+1]] else v: [0, 1, 1],

   for i from j thru 1 step(-1) do

    (h: vv[i], u: [v[1]+h*v[3], v[3], 2*h*v[1]+v[2]+h^2*v[3]], v: u),

   return(v)),

   for n from 1 thru nmax do

    (w: sqrt(n), m: floor(w),

     if w=m then  b: n else

      (v: coeff(w, m),  x: v[1], y: v[2], z: v[3], q: mod(z, 2),

       if q=0 then (z: z/2, y: y/2) else x: 2*x,

       fr: (x*m+y)/z, m: m+z, fr: fr+x, b: m^2+fr),

      printf( fl, "~d, ", b)),

      close(fl));

(Python)

from sympy import floor, S, sqrt

def coeff(w, m):

    a, p, s, vv = m, 0, w, []

    while a < 2*m:

        p += 1

        s = S.One/(s-floor(s))

        a = floor(s)

        if a < 2*m:

            vv.append(a)

    j = (p-1)//2

    v = [0, 1, 1] if p % 2 else [1, 0, vv[j]]

    for i in range(j-1, -1, -1):

        h = vv[i]

        v = [v[0]+h*v[2], v[2], 2*h*v[0]+v[1]+h**2*v[2]]

    return v

def A334116(n):

    w = sqrt(n)

    m = floor(w)

    if w == m:

        return n

    else:

        x, y, z = coeff(w, m)

        if z % 2:

            x *= 2

        else:

            z //= 2

            y //= 2

        return (m+z)**2+x+(x*m+y)//z # Chai Wah Wu, Sep 30 2021, after Maxima code

CROSSREFS

Cf. A002522, A005563, A002378, A013656, A147296, A154357, A033991, A007742, A157265, A154376, A154377, A156711, A156640, A154516, A154517, A154355, A157362.

Sequence in context: A011095 A020857 A096413 * A222591 A299447 A300085

Adjacent sequences:  A334113 A334114 A334115 * A334117 A334118 A334119

KEYWORD

nonn

AUTHOR

Gerhard Kirchner, Apr 14 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 17 14:19 EDT 2022. Contains 353746 sequences. (Running on oeis4.)