A078933 -id:A078933 - cp's OEIS frontend

A200218 The differences x^3 - y^2 of Danilov's subsequence of good Hall's examples A078933.

-297, 548147655, -1019827620252441, 1897387247823873407415, -3530085179800800999132960777, 6567716416847133270037051381858983, -12219223258107727669457593220846745613305, 22733840433256343397153666138928891468676446359

Offset: 1

Artur Jasinski, Nov 14 2011

For x values see A200216.
For y values see A200217.
All terms in this sequence are of the form: 3^3 * 11(2^3 * 31 * 61^2 * k + 922807).

Cf. A078933, A179107, A179108, A179109, A179387, A179388, A199496.

aa = {}; uu = 682 + 61 * Sqrt[125]; Do[vv = Expand[uu^(2 * n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5 * tt^2 - 3000 * tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, dd], {n, 1, 10}]; aa
(* Recurrence generator of R. J. Mathar *)
dd = {-297, 548147655, -1019827620252441}; a0 = dd[[1]]; a1 = dd[[2]]; a2 = dd[[3]]; Do[a = a0 + 1860497 * a1 - 1860497 * a2; a0 = a1; a1 = a2; a2 = a; AppendTo[aa, a], {n, 1, 10}]; aa
(* Third one after Lucas numbers formula *)
Table[27/125 (-5 + (-1)^n ((-1)^(n + 1) 6 + LucasL[15 (-1 + 2 n)])), {n, 10}] (* Artur Jasinski, Nov 18 2011*)

3125 * a(n)^2 + 6750 * a(n) + 729 = 2916 * A200216(n).
a(n) = (A200216(n))^3 - (A200217(n))^2.
Conjecture: a(n) = -1860497 * a(n-1) + 1860497 * a(n-2) + a(n-3) with g.f. 297 * z * (1 + 14882 * z + z^2) / ( (z-1)*(z^2 + 1860498 * z+1) ). - R. J. Mathar, Nov 15 2011
Hyperelliptic curve (157464*y)^2 = (729 + 594*d + 125*d^2) (-729 + 13500*d + 15625*d^2)^2 is singular (has two cusps) and for this reason Danilov's sequence has infinitely many integer solutions. - Artur Jasinski, Nov 16 2011
a(n) = (27/125) * (-5 + (-1)^n * ((-1)^(n+1) * 6 + L(15*(2*n - 1)))) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

A173348 Numbers x such that 0 < |x^7 - y^2| < x^(5/2) for some number y.

Original entry on oeis.org

12, 93, 239, 4896, 4904, 6546, 7806, 9104, 20542, 35962, 43783, 96569, 616400, 635331, 842163, 7888432, 450177181

Offset: 1

T. D. Noe, Feb 22 2010

Beukers and Stewart conjecture that for coprime integers n and m with n > m >= 2, and for any c > 0, the inequality 0 < |x^n - y^m| < c*X^(1-1/n-1/m) is true for infinitely many positive integers x and y, where X = max(x^n,y^m). They compute such x for 34 pairs (n,m). Given x, it is easy to compute y = round(x^(n/m)). Their tables have been extended to include all terms < 10^7 (or higher to obtain more terms).

a(18) > 10^9. - Robert Price, Apr 15 2021

F. Beukers and C. L. Stewart, Neighboring powers, J. Number Theory, 130 (2010), 660-679.

Cf. A078933 (m=2, n=3, Hall's conjecture)
Cf. A116884 (m=2, n=5)
This sequence (m=2, n=7)
Cf. A173349 (m=2, n=9)
Cf. A173350 (m=2, n=11)
Cf. A173351 (m=3, n=4)
Cf. A173352 (m=3, n=5)
Cf. A173353 (m=3, n=7)
Cf. A173354 (m=3, n=8)
Cf. A173355 (m=3, n=10)
Cf. A173356 (m=3, n=11)
Cf. A173357 (m=4, n=5)
Cf. A173358 (m=4, n=7)
Cf. A173359 (m=4, n=9)
Cf. A173360 (m=4, n=11)
Cf. A173361 (m=5, n=6)
Cf. A173362 (m=5, n=7)
Cf. A173363 (m=5, n=8)
Cf. A173364 (m=5, n=9)
Cf. A173365 (m=5, n=11)
Cf. A173366 (m=5, n=12)
Cf. A173367 (m=6, n=7)
Cf. A173368 (m=6, n=11)
Cf. A173369 (m=7, n=8)
Cf. A173370 (m=7, n=9)
Cf. A173371 (m=7, n=10)
Cf. A173372 (m=7, n=11)
Cf. A173373 (m=7, n=12)
Cf. A173374 (m=8, n=9)
Cf. A173375 (m=8, n=11)
Cf. A173376 (m=9, n=10)
Cf. A173377 (m=9, n=11)
Cf. A173378 (m=10, n=11)
Cf. A173379 (m=11, n=12)

Solutions[n_,m_,lim_] := Module[{x, y, t={}, pow=n*(1-1/m-1/n)}, Do[y=Round[x^(n/m)]; If[0 < Abs[x^n-y^m]

a(17) from Robert Price, Apr 15 2021

A179108 Values x for records of minima of positive distance d between a square cubefree integer y and a cube of positive and squarefree integer x and such d = y^2 - x^3.

Original entry on oeis.org

2, 46, 109, 5234, 8158, 720114, 28187351, 110781386, 154319269, 384242766, 390620082, 3790689201, 65589428378

Offset: 1

Artur Jasinski, Jun 29 2010

If x=n^2 and y=n^3 distance d=0.
For d values see A179107.
For y values see A179109.
For numbers x from 46 to 108 distance can't be less than 8.
For numbers x from 109 to 5233 distance can't be less than 15.
For numbers x from 5234 to 8157 distance can't be less than 17.
For numbers x from 8158 to 729113 distance can't be less than 24.
For numbers x from 729114 to 28187350 distance can't be less than 225.
Next conjectured terms are: 53197086958290, 12813608766102800, 810574762403977000, 471477085999389000.

Cf. A179107, A179109.

Cf. A078933. [From R. J. Mathar, Oct 13 2010]

d = 3; max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^d)^(1/2)] + 1; k = m^2 - n^d; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 720114}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx (* Artur Jasinski, Oct 30 2011 *)

A116884 Integers k such that 0 < |k^5 - m^2| <= k^(3/2) for some integer m.

Original entry on oeis.org

1, 5, 8, 23, 27, 55, 73, 76, 377, 396, 432, 18219, 18231, 747343, 748635, 5523608, 7626590, 32866452, 82251007, 1133553044, 1778903359, 3664408636, 7208605769, 26149894782

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

			|432^5 - 3878907^2| = 8217 < 432^(3/2).

Cf. A078933, A116885.

A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

Original entry on oeis.org

93844, 322001299796379844, 1114592308630995805123571151844, 3858108676488182444301031186675778188809844, 13354661111806898918013326915229994453818137920195953844

Offset: 1

Artur Jasinski, Nov 14 2011

nonn

For y values see A200217.
For x^3-y^2 values see A200218.
The values (a(n)+1)/5 are perfect squares: for sqrt((a(n)+1)/5) see A200335.
This sequence is an infinite subset of A078933. - Artur Jasinski, Nov 27 2011

			|93844^3 - (round(sqrt(93844^3)))^2| < sqrt(93844).

For references see A078933 and A179386.

L. V. Danilov, Letter to the Editor, Mat. Zametki, 1984, Volume 36, Issue 3, Pages 457-458.
L. V. Danilov, Letter to the editor, Math. Notes 36 (3) (1984) 726.
R. D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078 [math.NT], 2014.

Cf. A078933, A179107, A179108, A179109, A179387, A179388, A199496.

Cf. A200217, A200218, A200335.

aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, xx], {n, 1, 6}]; aa
(* second program follows the generator formula *)
data = Table[(7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2 n - 1)) - (1/f)^(15 (2 n - 1))) + (1/20)*(f^(30 (2 n - 1)) + (1/f)^(30 (2 n - 1)))) /. f -> GoldenRatio, {n, 1, 6}]; data // FunctionExpand // ExpandAll // Simplify (* Bob Hanlon (hanlonr(AT)cox.net) *)
(* third program uses the Lucas numbers formula *)
Table[7/10 + (-1)^(n + 1) 3/5 LucasL[15*(2 n - 1)] +
  1/20 LucasL[30*(2 n - 1)] , {n, 1, 10}] (* Artur Jasinski, Nov 18 2011 *)

u = quadunit(20)^5
for(i=1,6, v = u^(2*i-1); t = ((-1)^i * real(v) + 57) / 125; print(5^5*t^2 - 3000*t + 719) ) \\ Noam D. Elkies

from sympy import lucas
def A200216(n): return (14+12*(lucas(k:=30*n-15) if n&1 else -lucas(k:=30*n-15))+lucas(k<<1))//20 # Chai Wah Wu, Jun 19 2024

Conjecture: a(n) = 3461450947505*a(n-1) + 6440022564931157490*a(n-2) - 6440022564931157490*a(n-3) - 3461450947505*a(n-4) + a(n-5). - R. J. Mathar, Nov 15 2011
Conjecture: g.f. 4092*(1-z)/(5*(1+1860498*z+z^2)) - 7/(10*(z-1)) + 930249*(1-z)/(10*(1-3461452808002*z+z^2)). - R. J. Mathar, Nov 15 2011
3125*A200218(n)^2 + 6750*A200218(n) + 729 = 2916*a(n). - Artur Jasinski, Nov 15 2011
125*y^2 *(54 + 50*x^3 - 25*y^2)=(9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2. - Artur Jasinski, Nov 16 2011
a(n) = 7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2*n-1))-(1/f)^(15*(2*n-1))) + (1/20)*(f^(30*(2*n-1))+(1/f)^(30*(2*n-1))), where f is golden ratio constant = (1+sqrt(5)/2). - Artur Jasinski, Nov 17 2011
a(n) = 7/10 + (3/5)*L(15*(2*n - 1))*(-1)^(n+1) + (1/20)*L(30*(2*n - 1)) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

A200217 Danilov's sequence of y values satisfying 0 < |x^3 - y^2| < sqrt(x) with integer (x,y).

Original entry on oeis.org

28748141, 182720147509505842286585077, 1176722513851727970194784616032383058302343205, 7578123615032687003769196301877008424487234722410713810234126013

Offset: 1

Artur Jasinski, Nov 14 2011

nonn

See A078933 for further references.
All terms in this sequence are of the form 61*(2728*k + 2065).
Relations between y and x are given by the curve:
125*y^2 *(54 + 50*x^3 - 25*y^2) = (9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2.
Relations between y and d are given by the hyperelliptic curve:
(157464*y)^2 = (729 + 594*d + 125*d^2) (-729 + 13500*d + 15625*d^2)^2 is singular (has two cusps) and for these reasons Danilov's sequence has infinitely many integer solutions. - Artur Jasinski, Nov 16 2011

L. V. Danilov Letter to the editor, Math. Notes 36 (3) (1984) 726.

Cf. A078933, A200216 (x-values), A200218 (x^3-y^2), A179107 - A179109, A179387, A179388, A199496.

with(numtheory):
Di := 125 ;
cf := numtheory[cfrac](sqrt(Di),'periodic','quotients') ;
for i from 1 to 220 do
   x := nthnumer(cf,i) ;
   y := nthdenom(cf,i) ;
   rr := x^2-Di*y^2 ;
   if rr = -1 then
      t := x-5 ;
      if (t mod 5) = 2 then
              t := -t-10 ;
              y := -y ;
      end if;
      pk := t ;
      qk := y ;
      yM := qk*(pk^2+pk-1) ;
      yM := abs(yM) ;
      printf("%d,",yM) ;
   end if;
end do: # R. J. Mathar, Nov 15 2011

aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, yy], {n, 1, 5}]; aa
(* recurrence formula of R. J. Mathar *)
dd = {28748141, 182720147509505842286585077, 1176722513851727970194784616032383058302343205, 7578123615032687003769196301877008424487234722410713810234126013, 48803313311937248954865638168364942372153001387358275397822506563724900540813098269, 314294607902465331119210305427552029679173295887697814635011836442516313036620521777783545650437642949}; a0 = dd[[1]]; a1 = dd[[2]]; a2 = dd[[3]]; a3 = dd[[4]]; a4 = dd[[5]]; a5 = dd[[6]]; Do[a = 6440022564929296994 a5 + 22291834190970757443015664937985 a4 + -41473935220466903245533179036528718020 a3 + 22291834190970757443015664937985 a2 + 6440022564929296994*a1 - a0; a0 = a1; a1 = a2; a2 = a3 = a4; a5 = a6; a6 = a; AppendTo[aa, a], {n, 1, 10}]; aa (* Artur Jasinski, Nov 15 2011 *)
CoefficientList[Series[-(61 (-1 + x) (471281 - 39648020168249880312376 x - 417898575330317669831476343067314 x^2 - 39648020168249880312376 x^3 + 471281 x^4))/((1 - 6440026026380244498 x + x^2) (1 - 1860498 x + x^2) (1 + 3461452808002 x + x^2)), {x, 0, 10}], x] (* Artur Jasinski, Nov 16 2011 *)
(* Lucas - Fibonacci formula *)
aa = {}; Do[If[n == 1, AppendTo[aa, 15 + 9 LucasL[15 (-1 + 2 n)] + 15 LucasL[30 (-1 + 2 n)] - 6 Fibonacci[15 (-1 + 2 n)] + Fibonacci[30 (-1 + 2 n)]], AppendTo[aa, 15/8 Fibonacci[15 (-1 + 2 n)] - 9/20 Fibonacci[30 (-1 + 2 n)] + 1/40 Fibonacci[45 (-1 + 2 n)]]], {n, 1, 10}]; aa (* Artur Jasinski, Nov 18 2011 *)

Conjecture: a(n) = +6440022564929296994*a(n-1) +22291834190970757443015664937985*a(n-2) -41473935220466903245533179036528718020*a(n-3) +22291834190970757443015664937985*a(n-4) +6440022564929296994*a(n-5) -a(n-6). - R. J. Mathar, Nov 15 2011
Equivalent conjecture g.f.: -61*(z-1) * (471281*z^4 -39648020168249880312376*z^3 -417898575330317669831476343067314*z^2 -39648020168249880312376*z +471281) / ( (z^2+3461452808002*z+1) *(z^2-6440026026380244498*z+1) *(z^2-1860498*z+1) ). - R. J. Mathar, Nov 15 2011
Formula by Lucas and Fibonacci numbers: a(1) = 15+9*L(15)+15*L(30)-6*F(15)+F(30), for n>1 a(n) = (15/8)*F(15(2n-1)) - (9/20)*F(30(2n-1)) + (1/40) * F(45(2n-1)) where F(k) is k-th Fibonacci number A000045(n) and L(k) is k-th Lucas number A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

A199496 Good examples of a tightened Hall's conjecture: integers x such that 0 < x^3 - y^2 < sqrt(x) for some integer y.

Original entry on oeis.org

367806, 939787, 952764389446, 12438517260105, 35495694227489, 5853886516781223, 23415546067124892, 38115991067861271, 322001299796379844, 9870884617163518770, 42532374580189966073, 51698891432429706382, 601724682280310364065

Offset: 1

Artur Jasinski, Nov 07 2011

This sequence is the subset of x in A078933 such that x^3-(round(sqrt(x^3)))^2 is positive. This means the definition here does not take absolute values of x^3-y^2 as A078933 does.

Cf. A078933.

A116885 Integers n such that 0<|n^5-m^2|<= n for some integer m.

Original entry on oeis.org

1, 8, 55, 76, 377

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

Next term, if any, must be greater than 50*10^9.

			|377^5-2759646^2| = 341 <= 377.

Cf. A078933, A116884.

A199497 Distances d=x^3-y^2 for positive good examples of Hall's conjecture: positive integers d < sqrt(x).

Original entry on oeis.org

207, 307, 852135, 2767769, 5190544, 1641843, 105077952, 30032270, 548147655, 1651035656, 1878790553, 4101044247, 13027681441

Offset: 1

Artur Jasinski, Nov 07 2011

For values x see A199496.

For values y see A199498.

Cf. A078933, A199496, A199498.

A199498 Values y for positive good examples of Hall's conjecture: positive integers d = x^3-y^2 < sqrt(x).

Original entry on oeis.org

223063347, 911054064, 929989991784733049, 43868513629203032816, 211477180624706647625, 447884928428402042307918, 3583079427427216338463344, 7441505802879036345061579, 182720147509505842286585077, 31012309752051601656131750312, 277382747164996776244709473092

Offset: 1

Artur Jasinski, Nov 07 2011

nonn

For values x see A199496.

For values d see A199497.

Cf. A078933, A199496, A199497.

cp's OEIS Frontend

A200218 The differences x^3 - y^2 of Danilov's subsequence of good Hall's examples A078933.

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A173348 Numbers x such that 0 < |x^7 - y^2| < x^(5/2) for some number y.

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A179108 Values x for records of minima of positive distance d between a square cubefree integer y and a cube of positive and squarefree integer x and such d = y^2 - x^3.

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A116884 Integers k such that 0 < |k^5 - m^2| <= k^(3/2) for some integer m.

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A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

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A200217 Danilov's sequence of y values satisfying 0 < |x^3 - y^2| < sqrt(x) with integer (x,y).

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A199496 Good examples of a tightened Hall's conjecture: integers x such that 0 < x^3 - y^2 < sqrt(x) for some integer y.

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A116885 Integers n such that 0<|n^5-m^2|<= n for some integer m.

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A199497 Distances d=x^3-y^2 for positive good examples of Hall's conjecture: positive integers d < sqrt(x).

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A199498 Values y for positive good examples of Hall's conjecture: positive integers d = x^3-y^2 < sqrt(x).

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