A001014 -id:A001014 - cp's OEIS frontend

A179147 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 3 integral solutions.

343, 1331, 9261, 10648, 12167, 17576, 39304, 42875, 54872, 85184, 97336, 250047, 357911, 405224, 636056, 778688, 857375, 970299, 1331000, 1815848, 2146689, 2515456, 3511808, 3723875, 3944312, 4913000, 5359375, 5545233, 6128487, 6751269, 6859000, 7762392, 8120601, 8365427, 8869743, 9393931

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A054504, A081119, A179145-A179162.

Edited and extended by Ray Chandler, Jul 11 2010

a(30)-a(36) from Max Alekseyev, Jun 01 2023

A179151 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 7 integral solutions.

Original entry on oeis.org

8, 5832, 125000, 175616, 185193, 941192, 1404928, 1481544, 3241792, 4251528, 11239424, 11852352, 20346417, 21952000, 35937000, 37933056, 38614472, 48228544, 89915392, 128024064, 135005697, 193100552

Offset: 1

Artur Jasinski, Jun 30 2010

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A054504, A081119, A179145-A179162.

Edited and a(3)-a(22) from Ray Chandler, Jul 11 2010

A217982 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, vertical, diagonal or antidiagonal neighbors in a random 0..1 nXk array.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 6, 4, 4, 6, 10, 6, 16, 6, 10, 16, 10, 36, 36, 10, 16, 26, 16, 100, 98, 100, 16, 26, 42, 26, 256, 362, 362, 256, 26, 42, 68, 42, 676, 1180, 1942, 1180, 676, 42, 68, 110, 68, 1764, 4046, 8872, 8872, 4046, 1764, 68, 110, 178, 110, 4624, 13594, 43258, 54504

Offset: 1

R. H. Hardin Oct 16 2012

Table starts
...2...2.....4.......6........10.........16...........26............42
...2...2.....4.......6........10.........16...........26............42
...4...4....16......36.......100........256..........676..........1764
...6...6....36......98.......362.......1180.........4046.........13594
..10..10...100.....362......1942.......8872........43258........205446
..16..16...256....1180......8872......54504.......363728.......2347480
..26..26...676....4046.....43258.....363728......3375350......30097566
..42..42..1764...13594....205446....2347480.....30097566.....368335390
..68..68..4624...46052....986288...15361400....272859916....4595712084
.110.110.12100..155494...4714274...99957900...2457033362...56890420330
.178.178.31684..525730..22573862..651962620..22187715406..706581935362
.288.288.82944.1776548.108013892.4248252712.200128757472.8763874097740

			Some solutions for n=3 k=4
..0..0..0..0....0..0..1..1....0..0..0..1....1..1..1..1....1..1..0..0
..0..0..0..0....0..0..1..1....0..0..0..1....0..0..0..1....1..1..0..0
..1..1..1..1....0..0..1..1....1..1..1..1....0..0..0..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..287
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Columns 1 and 2 are A006355(n+1)

Column 3 is A206981(n-1)

A330482 Earliest start of a run of n numbers divisible by a sixth power larger than one.

Original entry on oeis.org

64, 16767, 26890623, 1507545109375, 777562026420218750, 283435321166212288109372

Offset: 1

Jud McCranie, Dec 16 2019

De Konnick's book gives probable terms a(5)=777562026420218750 and a(6)=283435321166212288109372.

			26890623 is divisible by 3^6, 26890624 is divisible by 2^6, and 26890625 is divisible by 5^6.  This is the smallest number with this property, so a(3) = 26890623.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Cf. A001014, A088080, A174113.

Cf. A045882, A271443, A330480, A330481, A330486, A330483, A330484, A330485.

a(5) from Giovanni Resta, Dec 17 2019

a(6) from Giovanni Resta, Dec 19 2019

A343286 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^6).

Original entry on oeis.org

1, 64, 729, 6176, 15625, 93312, 117649, 570048, 797526, 2000000, 1771561, 10474272, 4826809, 15059072, 22781250, 51231248, 24137569, 119066112, 47045881, 224500000, 171532242, 226759808, 148035889, 1085918400, 366218750, 617831552, 839677023, 1690380832, 594823321, 3645000000

Offset: 1

Ilya Gutkovskiy, Apr 10 2021

nonn

Cf. A001014, A001055, A023875, A050367, A329125, A343283, A343284, A343285, A343287, A343288, A343289.

A088017 Numbers not expressible as sum or difference of a nonzero cube and a nonzero square.

Original entry on oeis.org

6, 14, 16, 21, 27, 29, 32, 34, 42, 46, 51, 58, 59, 62, 66, 69, 70, 75, 77, 78, 84, 85, 86, 88, 90, 93, 96, 102, 103, 110, 111, 114, 115, 123, 125, 130, 133, 137, 140, 144, 149, 157, 158, 160, 162, 165, 166, 173, 176, 178, 179, 181, 182, 183, 187, 194, 201, 202, 203

Offset: 1

Hugo Pfoertner, Sep 18 2003

nonn

Numbers n such that neither variant of Mordell's equation y^2=x^3+n (A054504) or y^2=x^3-n (A081121) has an integral solution with nonzero x and y. - Jack Brennen, Aug 28 2003

			16 is in the sequence because the only integral solution to Mordell's equation y^2 = x^3 +- 16 is (y=4,x=0). 49 is not in the sequence because it can also be expressed as 65^3-524^2.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Eric Weisstein's World of Mathematics, Mordell Curve.

Cf. A054504, A081121, A087285, A087286.

A130555 Numbers that are sums of sixth powers of two distinct primes.

Original entry on oeis.org

793, 15689, 16354, 117713, 118378, 133274, 1771625, 1772290, 1787186, 1889210, 4826873, 4827538, 4842434, 4944458, 6598370, 24137633, 24138298, 24153194, 24255218, 25909130, 28964378, 47045945, 47046610, 47061506, 47163530

Offset: 1

Jonathan Vos Post, Aug 09 2007

This is to 6th powers as A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as sixth powers are cubes and the sum of cubes factorizations applies. There are semiprimes for values beginning a(1) = 793, a(2) = 15689 = 29 * 541, a(4) = 117713 = 53 * 2221, a(11) = 4826873 = 173 * 27901.

			a(1) = prime(1)^6 + prime(2)^6 = 2^6 + 3^6 = 64 + 729 = 793 = 13 * 61.

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Cf. A000040, A001014, A050997, A120398, A122616, A130873.

Select[Sort[Flatten[Table[Prime[n]^6 + Prime[k]^6, {n, 15}, {k, n - 1}]]], # <= Prime[15^6] &]
Union[Total/@Subsets[Prime[Range[20]]^6,{2}]] (* Harvey P. Dale, Mar 11 2012 *)

{A001014(A000040(i)) + A001014(A000040(j)) for i > j}.

A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 3, 2, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3

Offset: 1

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

nonn

a(n) = A081120(n)/2 if A081120(n) is even, (A081120(n)+1)/2 if A081120(n) is odd (i.e. if n is a cubic number).

Comment from T. D. Noe, Oct 12 2007: In sequences A134108 and A134109 (this entry) dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.

			y^2 = x^3 - 4 has solutions (y, x) = (2, 2) and (11, 5), hence a(4) = 2.
y^2 = x^3 - 5 has no solutions, hence a(5) = 0.
y^2 = x^3 - 8 has solution (y, x) = (0, 2), hence a(8) = 1.
y^2 = x^3 - 207 has 7 solutions (see A134106, A134107), hence a(207) = 7.

Jean-François Alcover, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A081120, A134106, A134107, A134108.

[ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];

A081120 = Cases[Import["https://oeis.org/A081120/b081120.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081120[[n]]}, If[EvenQ[an], an/2, (an+1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 28 2019 *)

A179136 Parameters n for which the elliptic curve y^2=x^3-n has rank 3.

Original entry on oeis.org

174, 307, 362, 431, 503, 516, 706, 713, 741, 755, 804, 984, 1048, 1075, 1173, 1187, 1192, 1208, 1236, 1259, 1315, 1356, 1439, 1478, 1588, 1607, 1668, 1712, 1724, 1727, 1763, 1777, 1812, 1902, 1951, 1966, 1999, 2001, 2036, 2071, 2181, 2188, 2198, 2219

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002150 (rank 0), A002152 (rank 1), A002154 (rank 2), A031508, A179137 (rank 4).

A179137 Parameters n for which the elliptic curve y^2=x^3-n has rank 4.

Original entry on oeis.org

2351, 3896, 4799, 4827, 5417, 5835, 6691, 6843, 9748, 9967, 10723, 11559, 12163, 12394, 12891, 13971, 14188, 14907, 15049, 15544

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002150 (rank 0), A002152 (rank 1), A002154 (rank 2), A179136 (rank 3).

Cf. A031508.

for(k=1, 1e4, if(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]==4, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

a(11)-a(20) from Seiichi Manyama, Jul 07 2019

cp's OEIS Frontend

A179147 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 3 integral solutions.

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A179151 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 7 integral solutions.

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A217982 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, vertical, diagonal or antidiagonal neighbors in a random 0..1 nXk array.

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A330482 Earliest start of a run of n numbers divisible by a sixth power larger than one.

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A343286 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^6).

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A088017 Numbers not expressible as sum or difference of a nonzero cube and a nonzero square.

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A130555 Numbers that are sums of sixth powers of two distinct primes.

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A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

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A179136 Parameters n for which the elliptic curve y^2=x^3-n has rank 3.

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A179137 Parameters n for which the elliptic curve y^2=x^3-n has rank 4.

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