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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

Table of n, a(n) for n=1..60.

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));

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

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Last modified May 14 16:45 EDT 2021. Contains 343888 sequences. (Running on oeis4.)