cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A200658 a(n) = A200656(n)^3 - A200657(n)^2.

Original entry on oeis.org

52488, 15336, -20088, 219375, -293625, 981504, -1285632, -474552, 1367631
Offset: 1

Views

Author

Artur Jasinski, Nov 20 2011

Keywords

Comments

For x values see A200656.
For y values see A200657.
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)*2^6, a(7)=a(3)*2^6.
A200218 is subset of this sequence.

Crossrefs

Cf. A200218, A200656, A200657.

A201048 Secondary terms in A200656.

Original entry on oeis.org

11512, 15448, 375376, 445528, 468472, 844596, 1002438, 1054062, 1782112, 1873888, 2346100, 2784550, 2927950, 3378384, 4009752, 4216248, 4598356, 4774960, 4993648, 57629500, 58652500, 61340526, 61827040, 62361952, 62786646, 63438544, 68412276, 70968942
Offset: 1

Views

Author

Artur Jasinski, Nov 26 2011

Keywords

Comments

For definition secondary terms see A200656.
For primary points in A200656 see A201047.
Secondary terms can be obtained from primary terms A201047 by ordinary multiplication by squares of particular integers while y is multiplicated by cube of that same integer and d by sixth power.
All terms in this sequence have square factor.

Links

  • Cf. A200656.

A077119 a(n) = A077118(n) - n^3.

Original entry on oeis.org

0, 0, 1, -2, 0, -4, 9, 18, 17, 0, 24, -35, 36, 12, -40, -11, 0, -13, -56, 30, -79, -45, -39, -67, 100, 0, 113, -83, -48, -53, -104, 138, -7, 163, -100, -26, 0, -28, -116, 217, 9, 248, -104, 17, 80, 79, 8, -139, 297, 0, 316, -155, 17, 119, 145, 89, -55
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Comments

a(n)=0 iff n = m^(6*k).
Values d=x^3-y^2 of extremal points of elliptic Mordell curves. Definition for extremal points see A200656. Each value x has only one value of distance d when coordinate x is extremal point, but for many fixed distances d, the elliptic curve has more than 1 extremal point. - Artur Jasinski, Nov 30 2011
Theorem (Artur Jasinski): If a(n)>0 then a(n)<(4n^(3/2)-1)/4 for every n. If a(n)<0 then a(n)>(-4n^(3/2)-1)/4 for every n. a(n)=0 then n is perfect square. - Artur Jasinski, Dec 08 2011

Examples

			A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.

Crossrefs

Cf. A000578, A077118, A077111.
|a(n)| = A002938(n).

Programs

  • Magma
    [Round(Sqrt(n^3))^2-n^3: n in [0..60]]; // Vincenzo Librandi, Mar 24 2015
  • Maple
    A077119 := proc(n)
    (round( sqrt(n^3) ))^2-n^3 ;
end proc: # R. J. Mathar, Jan 18 2021
  • Mathematica
    Table[Round[Sqrt[x^3]]^2 - x^3, {x, 0, 100}]  (* Artur Jasinski, Nov 30 2011 *)
  • Python
    from math import isqrt
def A077119(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2-k # Chai Wah Wu, Jul 29 2022

Formula

a(n) = if A077116(n) < A070929(n) then -A077116(n) else A070929(n).

A201047 Coordinates x of Mordell elliptic curves x^3-y^2 for primary extremal points with quadratic extensions over rationals.

Original entry on oeis.org

1942, 2878, 3862, 6100, 8380, 18694, 31228, 93844, 111382, 117118, 129910, 143950, 186145, 210025, 575800, 1193740, 1248412, 1326025, 1388545, 1501504, 1697908, 1813660, 1946737, 2069353, 2151262, 2305180, 3864190, 3897622, 54054144, 61974313, 63546025
Offset: 1

Views

Author

Artur Jasinski, Nov 26 2011

Keywords

Comments

For y coordinates see A201269.
For distances d between cubes and squares see A201268.
Primary points in A200656.
For definition primary points see A200656.
For secondary terms in A200656 see A201048.
For successive quadratic extensions see A201278.
Theorem (*Artur Jasinski*):
Every particular coordinate x contained only one extremal point.
Proof (*Artur Jasinski*): Coordinate y is computable from the formula y(x) = round(sqrt(x^3)) and distance d between cube of x and square of y is computable from the formula d(x) = x^3-(round(sqrt(x^3)))^2.

Crossrefs

Cf. A201268, A201269, A200656.

Formula

a(n) = (A201268(n)+(A201269(n))^2)^(1/3).

A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

Original entry on oeis.org

10, 2, 2, 5, 5, 130, 185, 5, 2, 2, 10, 10, 5, 5, 10, 17, 17, 5, 5, 5, 53, 53, 13, 13, 1490, 5, 2, 2, 5, 1565, 5
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2011

Keywords

Comments

Conjecture (Jasiński): The numbers in this sequence are multiplicative combinations of: primes congruent to 1 or 2 modulo 4 (A002313), Pythagorean primes (A002144), the number 2, and norms of Gaussian primes A055025.

Crossrefs

Cf. A200216, A200656, A201047.

Extensions

Minor edits by N. J. A. Sloane, Feb 23 2014

A200936 Successive values x of solutions Mordell's elliptic curve x^3-y^2 = d contained points {x,y} with quadratic extension sqrt(2) over rationals.

Original entry on oeis.org

22, 190, 2878, 3862, 111382, 117118, 3864190, 3897622, 131738902, 131933758, 4477986238, 4479121942, 152135692822, 152142312190, 5168228240638, 5168266821142, 175568164615702, 175568389479358, 5964152516784190, 5964153827385622, 202605635754466582
Offset: 0

Views

Author

Artur Jasinski, Nov 25 2011

Keywords

Comments

This sequence is equivalent of A200216, but A200216 was for quadratic field with extension sqrt(5).
Coefficient r=sqrt(x)/d tend to sqrt(2)/432 ~ 0.00327364 when x and d tend to infinity.
Starting from a(2)= 2878 all points are extremal (for definition see A200656).
(a(n)+10)/2 is perfect square of even number for each n.
All numbers in this sequence are of the form 2*(12*k+11).
For y values see A200937.
For d values see A200938.
When n is even d=A200938(n) are positive~, when n is odd d=A200938(n) are negative.

Examples

			a(3)=2878=A200656(1) because 2878^3-154396^2=15336.
G.f. = 22 + 190*x + 2868*x^2 + 3862*x^3 + 111382*x^4 + 117118*x^5 + ... - _Michael Somos_, Aug 23 2018

Links

Crossrefs

Cf. A001333, A200216, A200656.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*(11+84*z+904*z^2-2868*z^3+492*z^5 -12*z^7 +2266*z^4 -440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1)))); // G. C. Greubel, Jul 27 2018
  • Mathematica
    aa = {22, 190, 2878, 3862, 111382, 117118, 3864190, 3897622, 131738902}; a1 = aa[[1]]; a2 = aa[[3]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; a6 = aa[[6]]; a7 = aa[[7]]; a8 = aa[[8]]; a9 = aa[[9]]; Do[an = a9 + 40*a8 - 40*a7 - 206*a6 + 206*a5 + 40*a4 - 40*a3 - a2 + a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = a6; a6 = a7; a7 = a8; a8 = a9; a9 = an; AppendTo[aa, an], {nn, 20}]; aa
CoefficientList[Series[-2*(11 + 84*z + 904*z^2 - 2868*z^3 + 492*z^5 - 12*z^7 + 2266*z^4 - 440*z^6 + 11*z^8)/((z - 1) (z^2 + 6*z + 1) (1 - 6*z + z^2) (z^2 + 2*z - 1) (z^2 - 2*z - 1)), {z, 0, 30}], z] (* G. C. Greubel, Jul 27 2018 *)
a[ n_] := With[{m = Max[-5 - n, n]}, SeriesCoefficient[ 2 (1 - 12 x - 40 x^2 + 396 x^3 - 1138 x^4 + 396 x^5 - 40 x^6 - 12 x^7 + x^8) / (x^2 (x - 1) (1 + 6 x + x^2) (1 - 6 x + x^2) (x^2 + 2 x - 1) (x^2 - 2 x - 1)), {x, 0, m}]]; (* Michael Somos, Aug 23 2018 *)
a[ n_] := With[ {m = If[ OddQ[n], -5 - n, n], r1 = 1 + Sqrt[2], r2 = 1 - Sqrt[2]}, Simplify[7 - 6 (6 r1 + r2) r1^m - 6 (r1 + 6 r2) r2^m + (169 r1 + 29 r2)/4 r1^(2 m) + (29 r1 + 169 r2)/4 r2^(2 m)]]; (* Michael Somos, Aug 25 2018 *)
  • PARI
    z='z+O('z^30); Vec(2*(11+84*z+904*z^2-2868*z^3+492*z^5 -12*z^7 +2266*z^4 -440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1))) \\ G. C. Greubel, Jul 27 2018
  • PARI
    {a(n) = my(m = max(-5-n, n)); polcoeff( 2*(1 - 12*x - 40*x^2 + 396*x^3 - 1138*x^4 + 396*x^5 - 40*x^6 - 12*x^7 + x^8) / (x^2*(x - 1)*(1 + 6*x + x^2)*(1 - 6*x + x^2)*(x^2 + 2*x - 1)*(x^2 - 2*x - 1)) + x * O(x^m), m)}; /* Michael Somos, Aug 23 2018 */
  • PARI
    {a(n) = my(m = if(n%2, -5-n, n), r1 = 1 + quadgen(8), r2 = 1 - quadgen(8)); simplify(7 - 6*(6*r1 + r2) * r1^m - 6*(r1 + 6*r2) * r2^m + (169*r1 + 29*r2)/4 * r1^(2*m) + (29*r1 + 169*r2)/4 * r2^(2*m))}; /* Michael Somos, Aug 25 2018 */

Formula

a(n) = (A200937(n)^2 + A200938(n))^(1/3).
a(n) = a(n-1)+ 40*a(n-2) - 40*a(n-3) - 206*a(n-4) + 206*a(n-5) + 40*a(n-6) - 40*a(n-7) - a(n-8) + a(n-9).
G.f.: 2*(11+84*z+904*z^2-2868*z^3+492*z^5-12*z^7+2266*z^4-440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1)).
a(2*n + 1) - a(2*n) = 24 * A001333(2*n + 3), a(n) = a(-5-n) for all n in Z. - Michael Somos, Aug 23 2018

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

Views

Author

Artur Jasinski, Nov 20 2011

Keywords

Comments

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.

Crossrefs

Cf. A200217, A200656, A200658.

A201225 Values x for infinite sequence x^3-y^2 = d with decreasing coefficient r=sqrt(x)/d which tend to 1/(1350*sqrt(5))or infinity family of solutions Mordell curve with extension sqrt(5).

Original entry on oeis.org

6100, 2305180, 748476100, 241118603980, 77641444770100, 25000340035616380, 8050032494909496100, 2592085474592828222380, 834643472994047002110100, 268752606222334691877221980, 86537504560185639786707316100, 27864807715774753485364243735180
Offset: 1

Views

Author

Artur Jasinski, Nov 28 2011

Keywords

Comments

a(1) = A200656(4) = A201047(4).
a(2) = A200656(36) = A201047(26).
All points in this sequence are extremal points (definition see A200656) and from these reason is subset of A200656 and primary (definition see A200656) and from these reason is subset of A201047.

Links

Crossrefs

Cf. A200216, A200656.

Programs

  • Mathematica
    LinearRecurrence[{341,-6138,6138,-341,1},{6100,2305180,748476100,241118603980,77641444770100},20] (* Harvey P. Dale, Aug 17 2016 *)

Formula

G.f.: (20*(-305-11254*z+7424*z^2-346*z^3+z^4))/((-1+z)*(1- 322*z+z^2)*(1-18*z+z^2)).
a(n) = 341*a(n-1) - 6138*a(n-2) + 6138*a(n-3) - 341*a(n-4) + a(n-5).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.