A001014 -id:A001014 - cp's OEIS frontend

A022522 Nexus numbers (n+1)^6 - n^6.

1, 63, 665, 3367, 11529, 31031, 70993, 144495, 269297, 468559, 771561, 1214423, 1840825, 2702727, 3861089, 5386591, 7360353, 9874655, 13033657, 16954119, 21766121, 27613783, 34655985, 43067087, 53037649, 64775151, 78504713, 94469815, 112933017, 134176679

Offset: 0

N. J. A. Sloane, Jun 14 1998

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Nexus Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Column k=5 of array A047969.
Beginning with n=1, a subsequence of A181125 (difference of two positive 6th powers). - Mathew Englander, Jun 01 2014
Cf. A008292, A022521, A022523.

List([0..30], n-> (n+1)^6 - n^6); # G. C. Greubel, Aug 28 2019

[(n+1)^6-n^6: n in [0..30]]; // Vincenzo Librandi, Nov 22 2011

A022522:=n->(n + 1)^6 - n^6; seq(A022522(n), n=0..30); # Wesley Ivan Hurt, Jan 30 2014

Table[(n+1)^6-n^6, {n,0,30}] (* Vincenzo Librandi, Nov 22 2011 *)

a(n)=(n+1)^6-n^6 \\ Charles R Greathouse IV, Sep 24 2015

[(n+1)^6 -n^6 for n in (0..30)] # G. C. Greubel, Aug 28 2019

G.f.: (1+x)*(1+56*x+246*x^2+56*x^3+x^4)/(1-x)^6. - Colin Barker, Dec 21 2012
a(n) = A005408(n) * A243201(n). - Mathew Englander, Jun 06 2014
a(n) = A001014(n+1) - A001014(n). - Wesley Ivan Hurt, Jun 06 2014
E.g.f.: (1 +62*x +270*x^2 +260*x^3 +75*x^4 +6*x^5)*exp(x). - G. C. Greubel, Aug 28 2019
G.f.: polylog(-6, x)*(1-x)/x. See the g.f. of Colin Barker (with expanded numerator), and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

More terms from Colin Barker, Dec 21 2012

A007412 The noncubes: a(n) = n + floor((n + floor(n^(1/3)))^(1/3)).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71

Offset: 1

N. J. A. Sloane

Seems to be numbers k for which the order of the torsion subgroup t of the elliptic curve y^2 = x^3 - k is t=1. - Artur Jasinski, Jun 30 2010

A010057(a(n)) = 0. - Reinhard Zumkeller, Oct 22 2011

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 27911
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. J. dos Reis and D. M. Silberger, Generating nonpowers by formula, Math. Mag., 63 (1990), 53-55.
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]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
R. D. Nelson, Sequences which omit powers, The Mathematical Gazette, Number 461, 1988, pages 208-211.

Cf. A000578 (complement), A000037 (nonsquares).

a007412 n = n + a048766 (n + a048766 n)  -- Reinhard Zumkeller, Oct 22 2011

With[{upto=58},Complement[Range[upto],Range[Ceiling[Power[upto, (3)^-1]]]^3]] (* Harvey P. Dale, Nov 09 2011 *)
A007412Q = ! IntegerQ[#~Surd~3] &; Select[Range[57], A007412Q] (* JungHwan Min, Mar 27 2017 *)

lista(nn) = for (n=1, nn, if (! ispower(n, 3), print1(n, ", "))); \\ Michel Marcus, May 24 2015

list(lim)=my(v=List(),s=sqrtnint(lim\=1,3),k3,k13=1); for(k=1,s, k3=k13; k13=(k+1)^3; for(n=k3+1,k13-1, listput(v,n))); for(n=s^3+1,lim, listput(v,n)); Vec(v) \\ Charles R Greathouse IV, Jun 13 2024

from sympy import integer_nthroot
def A007412(n): return n+(k:=integer_nthroot(n,3)[0])+int(n>=(k+1)**3-k) # Chai Wah Wu, Jun 17 2024

a(n) = n + A048766(n + A048766(n)). - Reinhard Zumkeller, Oct 22 2011

a(n) = n + n^(1/3) + O(1). - Charles R Greathouse IV, Aug 08 2024

A368714 Numbers whose maximal exponent in their prime factorization is even.

Original entry on oeis.org

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208

Offset: 1

Amiram Eldar, Jan 04 2024

First differs from A240112 at n = 30.
Numbers k such that A051903(k) is even.
The asymptotic density of this sequence is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A240112.
Subsequences: A000290, A000302, A000583, A001014, A001016, A001025, A008454, A062503, A067259.
Similar sequences: A060476, A074661, A096432, A336064, A368715.

Select[Range[210], # == 1 || EvenQ[Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || !(vecmax(factor(k)[,2])%2), print1(k, ", ")));

A179162 a(n) = least positive k such that Mordell's equation y^2 = x^3 + k has exactly n integral solutions.

Original entry on oeis.org

6, 27, 2, 343, 12, 1, 37, 8, 24, 512, 9, 35611289, 73, 10218313, 315, 129554216, 17, 274625, 297, 17576000, 2817, 200201625, 1737

Offset: 0

Artur Jasinski, Jun 30 2010

Additional known terms: a(24)=4481, a(26)=225, a(28)=2089, a(32)=1025.
For least positive k such that equation y^2 = x^3 - k has exactly n integral solutions, see A179175.
If n is odd, then a(n) is perfect cube. [Ray Chandler]

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(11), a(13), a(15), a(17), a(19), a(21) added by Ray Chandler, Jul 11 2010

A179163 Numbers k such that Mordell's equation y^2 = x^3 - k has exactly 1 integral solution.

Original entry on oeis.org

1, 8, 27, 64, 125, 512, 729, 1000, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 15625, 19683, 24389, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 59319, 64000, 68921, 79507, 91125, 97336, 110592, 117649, 125000, 132651

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

Contains all sixth powers: suppose that y^2 = x^3 - t^6, then (y/t^3)^2 = (x/t^2)^3 - 1. The elliptic curve Y^2 = X^3 - 1 has rank 0 and the only rational points on it are (1,0), so y^2 = x^3 - t^6 has only one solution (t^2,0). - Jianing Song, Aug 24 2022

Jianing Song, Table of n, a(n) for n = 1..108 (using data from A179149)
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. A081120, A081121, A179163-A179175, A356713.
Complement of A179149 among the positive cubes.
Cf. also A179145, A356703.

(* Assuming every term is a cube *) xmax = 2000; r[n_] := Reap[Do[rpos = Reduce[y^2 == x^3 - n, y, Integers]; If[rpos =!= False, Sow[rpos]]; rneg = Reduce[y^2 == (-x)^3 - n, y, Integers]; If[rneg =!= False, Sow[rneg]], {x, 1, xmax}]]; ok[1] = True; ok[n_] := Which[rn = r[n]; rn[[2]] === {}, False, Length[rn[[2]]] > 1, False, ! FreeQ[rn[[2, 1]], Or], False, True, True]; ok[n_ /; !IntegerQ[n^(1/3)]] = False; A179163 = Reap[Do[If[ok[n], Print[n]; Sow[n]], {n, 1, 140000}]][[2, 1]] (* Jean-François Alcover, Apr 12 2012 *)

a(n) = A356713(n)^3. - Jianing Song, Aug 24 2022

Edited and extended by Ray Chandler, Jul 11 2010

A060950 Rank of elliptic curve y^2 = x^3 + n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 10 2001

The curves for n and -27*n are isogenous (as Noam Elkies points out--see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013

			a(1) = A060951(27) = a(729) = 0. - _Jonathan Sondow_, Sep 10 2013

T. D. Noe, Table of n, a(n) for n = 1..10000 (from Gebel)
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]
H. Mishima, Tables of Elliptic Curves
T. Womack, Minimal-known positive and negative k for Mordell curves of given rank

Cf. A031507, A002151, A002153, A002155, A102833, A179124.
Cf. A060748, A060838, A060951, A060952, A060953.
Cf. A081119 (number of integral solutions to Mordell's equation y^2 = x^3 + n).

a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, n]))[1] \\ Jianing Song, Aug 24 2022

apply( {A060950(n)=ellrank(ellinit([0, n]))[1]}, [1..99]) \\ For PARI version  < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024

a(n) = A060951(27*n) and A060951(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013

Corrected by James R. Buddenhagen, Feb 18 2005

A002155 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 2.

Original entry on oeis.org

15, 17, 24, 37, 43, 57, 63, 65, 73, 79, 89, 101, 106, 122, 129, 131, 142, 145, 148, 151, 161, 164, 168, 171, 186, 195, 197, 198, 204, 217, 222, 223, 225, 229, 232, 233, 248, 252, 260, 265, 268, 269, 281, 294, 295, 297, 303, 322, 331, 337, 347, 350, 353, 360, 366, 369, 373, 377, 381, 388, 389, 392, 404, 409, 412, 414, 433, 449, 464, 469, 481, 483, 485, 492

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1724 (using Gebel)
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
L. Lehman, Elliptic Curves of Rank Two. [broken link]
H. Mishima, Tables of Elliptic Curves.

Cf. A060950, A002151, A002153, A102833, A060748, A060838, A060951-A060953.

for k in[1..500] do if Rank(EllipticCurve([0,0,0,0,k])) eq 2 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019

More terms from James R. Buddenhagen, Feb 18 2005

A179175 a(n) = least positive k such that Mordell's equation y^2 = x^3 - k has exactly n integral solutions.

Original entry on oeis.org

3, 1, 2, 1331, 4, 216, 28, 54872, 116, 343, 828, 250047, 496, 71991296, 207

Offset: 0

Artur Jasinski, Jun 30 2010

The status of further terms is:
  15 integral solutions: unknown
  16 integral solutions: 503
  17 integral solutions: unknown
  18 integral solutions: 431
  19 integral solutions: unknown
  20 integral solutions: 2351
  21 integral solutions: unknown
  22 integral solutions: 3807
For least positive k such that equation y^2 = x^3 + k has exactly n integral solutions, see A179162.
If n is odd, then a(n) is perfect cube. [Ray Chandler]
From Jose Aranda, Aug 04 2024: (Start)
About those unknown terms:
a(15) <=    2600^3 = (26* 10^2)^3
a(17) <=   10400^3 = (26* 20^2)^3
a(19) <=   93600^3 = (26* 60^2)^3
a(21) <= 4586400^3 = (26*420^2)^3
The term a(13) = 71991296 = 416^3 = (26*4^2)^3. (End)

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. A081120, A081121, A179163-A179174.

Edited and a(7), a(11), a(13) added by Ray Chandler, Jul 11 2010

A256840 Primes of form n^2 + 20736.

Original entry on oeis.org

20857, 21577, 21961, 23761, 27961, 28657, 29017, 29761, 30937, 33961, 34897, 37897, 41761, 42937, 49297, 51361, 60337, 62761, 65257, 80761, 83737, 93097, 107761, 111337, 113761, 122497, 132961, 142537, 151057, 164377, 173617, 181537, 188017, 192961, 218761

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, 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]

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256841 (b=13).

a256840 n = a256840_list !! (n-1)
a256840_list = [x | x <- map (+ 20736) a000290_list, a010051' x == 1]

A002151 Numbers k for which rank of the elliptic curve y^2 = x^3 + k is 0.

Original entry on oeis.org

1, 4, 6, 7, 13, 14, 16, 20, 21, 23, 25, 27, 29, 32, 34, 42, 45, 49, 51, 53, 59, 60, 64, 70, 75, 78, 81, 84, 85, 86, 87, 88, 90, 93, 95, 96, 104, 109, 114, 115, 116, 123, 124, 125, 135, 137, 140, 144, 153, 157, 158, 159, 160, 162, 165, 167, 173, 175, 176, 178

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..2907 (using Gebel)
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
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]
L. Lehman, Elliptic Curves of Rank Zero.
H. Mishima, Tables of Elliptic Curves.

Cf. A060950, A002153, A002155, A102833, A060748, A060838, A060951, A060952, A060953.

for k in[1..200] do if Rank(EllipticCurve([0,0,0,0,k])) eq 0 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019

Corrected and extended by James R. Buddenhagen, Feb 18 2005

The missing entry 123 was added by T. D. Noe, Jul 24 2007

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A022522 Nexus numbers (n+1)^6 - n^6.

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A007412 The noncubes: a(n) = n + floor((n + floor(n^(1/3)))^(1/3)).

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A368714 Numbers whose maximal exponent in their prime factorization is even.

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A179162 a(n) = least positive k such that Mordell's equation y^2 = x^3 + k has exactly n integral solutions.

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A179163 Numbers k such that Mordell's equation y^2 = x^3 - k has exactly 1 integral solution.

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A060950 Rank of elliptic curve y^2 = x^3 + n.

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A002155 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 2.

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