A200217 -id:A200217 - cp's OEIS frontend

A200526 a(n) = gcd(t(n), t(3n-1)), where t = A200217.

2156316023, 211148507797805, 392841376460687116573, 13886731309220741899538675431, 1359801885649216204023955447726829, 2529908688645864568558938918274367865293, 89430911052730984787593270943984417274689212615

Offset: 2

Artur Jasinski, Nov 18 2011

nonn

Successive maxima of the GCD in A200217 occur between A200217(n) and A200217(3n-1) terms. Conjecture: All terms have same set of prime divisors, that can be used to complete prime factorization of every term in this sequence by the GCD method. All prime divisors with exception 19 are of the form 4k+1. The integer 19 divides a(3n+1) for n=0,1,2,3,...

Cf. A200217.

ff = {}; Do[AppendTo[ff, GCD[15/8 Fibonacci[15 (-1 + 2 n)] - 9/20 Fibonacci[30 (-1 + 2 n)] + 1/40 Fibonacci[45 (-1 + 2 n)], 15/8 Fibonacci[15 (-1 + 2 (3 n - 1))] - 9/20 Fibonacci[30 (-1 + 2 (3 n - 1))] + 1/40 Fibonacci[45 (-1 + 2 (3 n - 1))]]], {n, 2, 10}]; ff

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

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

Original entry on oeis.org

-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

A200657 Successive values y such that Mordell elliptic curve x^3 - y^2 = d has a quadratic extension over rationals.

Original entry on oeis.org

85580, 154396, 240004, 476425, 767125, 1235168, 1920032, 2555956, 5518439

Offset: 1

Artur Jasinski, Nov 20 2011

For x values see A200656.
For d values see A200658.
Definition: Secondary terms occurred when existed such integer k that A200656 is divisible by k^2 and A200657 is divisible by k^3 and A200658 is divisible by k^6.
Terms free of such k are primary terms.
Secondary terms are: a(6)=a(2)*2^3, a(7)=a(3)*2^3.
A200217 is subset of this sequence.

Cf. A200217, A200656, A200658.

A200937 Values y for infinite sequence x^3 - y^2 = d with increasing coefficient r = sqrt(x)/|d| or family of solutions Mordell curve with extension sqrt(2).

Original entry on oeis.org

100, 2620, 154396, 240004, 37172564, 40080716, 7596048140, 7694839700, 1512067083076, 1515423087964, 299656796131324, 299770801505956, 59339881525800500, 59343754352533100, 11749314454296080876, 11749446016399614644, 2326315710145219660324, 2326320179383913075836, 460599127771776655165660, 460599279594330127759300

Offset: 0

Artur Jasinski, Nov 25 2011

nonn

For x values see A200936.
For d values see A200938.
This sequence is equivalent of A200217, but A200217 was for quadratic field with extension sqrt(5).
All numbers in this sequence are of the form 4*(2*n+1).

G. C. Greubel, Table of n, a(n) for n = 0..850
Index entries for linear recurrences with constant coefficients, signature (0,238,0,-8127,0,40868,0,-8127,0, 238,0,-1).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((100 +2620*x +130596*x^2 -383556*x^3 +1239016*x^4 +4252504*x^5 -332600*x^6 -932360*x^7 +10356*x^8 +27564*x^9 -44*x^10 -116*x^11)/( (x^2+6*x+1)*(x^2-6*x+1)*(x^2+2*x-1)*(x^2-2*x-1)*(x^2+14*x-1)*(x^2 -14*x -1)))); // G. C. Greubel, Aug 22 2018

aa = {100, 2620, 154396, 240004, 37172564, 40080716, 7596048140, 7694839700, 1512067083076, 1515423087964, 299656796131324, 299770801505956}; a1 = aa[[1]]; a2 = aa[[2]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; a6 = aa[[6]]; a7 = aa[[7]]; a8 = aa[[8]]; a9 = aa[[9]]; a10 = aa[[10]]; a11 = aa[[11]]; a12 = aa[[12]]; Do[an = 238*a11 - 8127*a9 + 40868*a7 - 8127*a5 + 238*a3 - a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = a6; a6 = a7; a7 = a8; a8 = a9; a9 = a10; a10 = a11; a11 = a12; a12 = an; AppendTo[aa, an], {nn, 1, 88}]; aa
LinearRecurrence[{0, 238, 0, -8127, 0, 40868, 0, -8127, 0, 238, 0, -1}, {100, 2620, 154396, 240004, 37172564, 40080716, 7596048140, 7694839700, 1512067083076, 1515423087964, 299656796131324, 299770801505956}, 50] (* G. C. Greubel, Aug 22 2018 *)

x='x+O('x^30); Vec((100 +2620*x +130596*x^2 -383556*x^3 +1239016*x^4 +4252504*x^5 -332600*x^6 -932360*x^7 +10356*x^8 +27564*x^9 -44*x^10 -116*x^11)/( (x^2+6*x+1)*(x^2-6*x+1)*(x^2+2*x-1)*(x^2-2*x-1)*(x^2+14*x-1)*(x^2 -14*x -1))) \\ G. C. Greubel, Aug 18 2018

a(n) = sqrt(A200936(n)^3 - A200938(n)).
a(n) = 238*a(n-2) - 8127*a(n-4) + 40868*a(n-6) - 8127*a(n-8) + 238*a(n-10) - a(n-12).
G.f.: (100 + 2620*x + 130596*x^2 - 383556*x^3 + 1239016*x^4 + 4252504*x^5 - 332600*x^6 - 932360*x^7 + 10356*x^8 + 27564*x^9 - 44*x^10 - 116*x^11)/( (x^2+6*x+1)*(x^2-6*x+1)*(x^2+2*x-1)*(x^2-2*x-1)*(x^2+14*x-1)*(x^2 - 14*x - 1)). - R. J. Mathar, Nov 25 2011

Data corrected by G. C. Greubel, Aug 22 2018

A200938 Values d for infinite sequence x^3-y^2 = d with increasing coefficient r=sqrt(x)/|d| or family of solutions Mordell curve with extension sqrt(2).

Original entry on oeis.org

648, -5400, 15336, -20088, 100872, -105624, 599400, -604152, 3505032, -3509784, 20440296, -20445048, 119146248, -119151000, 694446696, -694451448, 4047543432, -4047548184, 23590823400, -23590828152, 137497406472, -137497411224, 801393624936, -801393629688

Offset: 0

Artur Jasinski, Nov 25 2011

For x values see A200936.
For y values see A200937.
This sequence is equivalent of A200218, but A200218 was for quadratic field with extension sqrt(5).
All numbers in this sequence are of the form 216*(4k+3).
When indices n are even d=a(n) are positive, when n is odd  d=a(n) are negative.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

Cf. A200216, A200217, A200218, A200936, A200937.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(216*(3-28*x+78*x^2+4*x^3-13*x^4)/((1-x)*(1+2*x-x^2)*(1-2*x-x^2)))); // G. C. Greubel, Aug 18 2018

uu = {648, -5400, 15336, -20088, 100872}; a1 = aa[[1]]; a2 = aa[[2]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; Do[an = a5 + 6 a4 - 6 a3 - a2 + a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = an; AppendTo[uu, an], {nn, 1, 20}]; uu

my(x='x+O('x^30)); Vec(216*(3-28*x+78*x^2+4*x^3-13*x^4)/((1-x)*(1+2*x-x^2)*(1-2*x-x^2))) \\ G. C. Greubel, Aug 18 2018

a(n) = A200936(n)^3 - A200937(n)^2.
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
G.f.: 216*(3 - 28*z + 78*z^2 + 4*z^3 - 13*z^4)/((1 - z)*(1 + 2*z - z^2) *(1 - 2*z - z^2)).
E.g.f.: 216*(cosh(x)*(14*cosh(sqrt(2)*x) - 4*sqrt(2)*sinh(sqrt(2)*x) - 11) - sinh(x)*(6*cosh(sqrt(2)*x) - 10*sqrt(2)*sinh(sqrt(2)*x) + 11)). - Stefano Spezia, Oct 03 2022

A200335 a(n) = sqrt((A200216(n)+1)/5).

Original entry on oeis.org

137, 253772063, 472142416783537, 878420022140682133063, 1634298694352222684783778137, 3040609452244043180572708973082863, 5657047804679503550674811676317937783937, 10524926126507566387571141730985597902165021463

Offset: 1

Artur Jasinski, Nov 16 2011

nonn

All numbers (A200216(n)+1)/5 are perfect squares

Vincenzo Librandi, Table of n, a(n) for n = 1..50
Index entries for linear recurrences with constant coefficients, signature (1860497, 1860497, -1).

Cf. A200216, A200217, A200218.

I:=[137, 253772063, 472142416783537]; [n le 3 select I[n] else 1860497*Self(n-1)+1860497*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Nov 18 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, Sqrt[(xx + 1)/5]], {n, 1, 20}]; aa

x='x+O('x^30); Vec((137 -1116026*x +137*x^2)/(1 - 1860497*x - 1860497*x^2 + x^3)) \\ G. C. Greubel, Jul 10 2018

G.f.: (137 - 1116026*x + 137*x^2)/(1 - 1860497*x - 1860497*x^2 + x^3).

a(n) = 1860497*a(n-1) + 1860497*a(n-2) - a(n-3). [corrected by Vincenzo Librandi, Nov 18 2011]

A201269 Coordinates y of points {x,y} of Mordell elliptic curves x^3-y^2 for primary extremal points with quadratic extensions over rationals.

Original entry on oeis.org

85580, 154396, 240004, 476425, 767125, 2555956, 5518439, 28748141, 37172564, 40080716, 46823500, 54615700, 80311375, 96251275, 436925600, 1304261335, 1394880175, 1526959675, 1636213375, 1839881024, 2212438625, 2442495725, 2716194871, 2976815179, 3155294924

Offset: 1

Artur Jasinski, Nov 29 2011

nonn

For x coordinates see A201047.
For distances d between cubes and squares see A201268.
For successive quadratic extensions see A201278.
Theorem (*Artur Jasinski*):
Every particular coordinate y contained only one extremal point.
Proof (*Artur Jasinski*): Coordinate x is computable from the formula x(y) = round(y^(2/3)) and distance d between cube of x and square of y is computable from the formula d(y) = round(y^(2/3))^3-y^2.

Cf. A200217, A201047, A201268.

a(n) = sqrt(A201047(n)^3-A201268(n)).

A200565 Integral x solutions of elliptic curve x^3-y^2 = 54814765 = A200218(2).

Original entry on oeis.org

819, 5256, 838044, 322001299796379844

Offset: 1

Artur Jasinski, Nov 19 2011

a(4)=A200216(2).

Cf. A200216, A200217, A200218, A134105.

A200918 Successive prime factors of (3^1006003 - 3)/1006003^2.

Original entry on oeis.org

2, 2, 2, 3, 7, 13, 19, 37, 757, 111779, 670669, 6371347, 34204069, 166437443, 310854619, 385634101, 14188652209, 42594124681, 10825536799379161, 154680726732318637

Offset: 1

Artur Jasinski, Nov 24 2011

1006003 = A014127(2).
2 is the only prime that occurs to a power greater than 1.
Conjecture (*Artur Jasinski*): If another infinite sequences with good Hall's examples occurred, it would have to contain primes from this sequence as constant divisors of the whole sequence, because parts of Danilov's infinite sequence (A200216, A200217, A200218) contain divisors of (3^A014127(1) - 3)/(A014127(1)^2).
a(21) > 10^18. - Max Alekseyev, Feb 26 2020

FactorDB, Factorization of 3^1006002 - 1.

Cf. A014127.

More terms from Sean A. Irvine, Sep 06 2012

a(19)-a(20) from Max Alekseyev, Feb 13 2020

cp's OEIS Frontend

A200526 a(n) = gcd(t(n), t(3n-1)), where t = A200217.

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

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A200657 Successive values y such that Mordell elliptic curve x^3 - y^2 = d has a quadratic extension over rationals.

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A200937 Values y for infinite sequence x^3 - y^2 = d with increasing coefficient r = sqrt(x)/|d| or family of solutions Mordell curve with extension sqrt(2).

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A200938 Values d for infinite sequence x^3-y^2 = d with increasing coefficient r=sqrt(x)/|d| or family of solutions Mordell curve with extension sqrt(2).

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A200335 a(n) = sqrt((A200216(n)+1)/5).

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A201269 Coordinates y of points {x,y} of Mordell elliptic curves x^3-y^2 for primary extremal points with quadratic extensions over rationals.

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