A179388 -id:A179388 - cp's OEIS frontend

A179791 Values x for records of the minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).

2, 3, 5, 6, 8, 13, 22, 23, 27, 62, 78, 147, 181, 203, 233, 468, 892, 1110, 1827, 3657, 3723, 10637, 11145, 11478, 12275, 16764, 19151, 22719, 23580, 24974, 30163, 36885, 41759, 41948, 44427, 66443, 86167, 96658, 115992, 222962, 248461, 248588, 384573

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^9.
For d values see A179790.
For y values see A179792.
Conjecture (Artur Jasinski): For any positive number x >= A179791(n), the distance d between the ninth power of x and the square of any y (such that x <> k^2 and y <> k^9) can't be less than A179790(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.

d = 9; 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)]; k = n^d - m^2; 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, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx

A179792 Values y for records of the minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).

Original entry on oeis.org

22, 140, 1397, 3174, 11585, 102978, 1098758, 1342070, 2761448, 116348986, 326908123, 5661454305, 14439547606, 24195364585, 44988513611, 1037782490126, 18907836782131, 50577039498042, 476237361126871, 10815891488601655

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^9.
For d values see A179790.
For x values see A179791.
Conjecture (Artur Jasinski): For any positive number x >= A179791(n), the distance d between the ninth power of x and the square of any y (such that x <> k^2 and y <> k^9) can't be less than A179790(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.

d = 9; 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)]; k = n^d - m^2; 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, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy

A179793 Records of minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).

Original entry on oeis.org

23, 747, 8847, 12654, 166831, 484471, 573055, 1248668, 1602775, 8764352, 72820023, 94338007, 143404871, 155195023, 262310000, 1529935249, 4884962400, 19571071932, 146228748359, 318603821009, 635586109888, 1305633968055

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^11.
For x values see A179794.
For x values see A179795.
Conjecture (Artur Jasinski): For any positive number x >= A179794(n), the distance d between the eleventh power of x and the square of any y (such that x <> k^2 and y <> k^11) can't be less than A179793(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.

d = 11; 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)]; k = n^d - m^2; 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, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd

A179794 Values x for records of the minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).

Original entry on oeis.org

2, 3, 6, 7, 8, 10, 14, 18, 20, 26, 28, 32, 38, 52, 60, 77, 145, 168, 222, 237, 268, 279, 286, 359, 367, 390, 536, 569, 622, 872, 1085, 1349, 1462, 1760, 1932, 2423, 2801, 5559, 5925, 7052, 8383, 8752, 10075, 11917, 15712, 17420, 17598, 23712, 26026, 28095

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^11.
For x values see A179794.
For x values see A179795.
Conjecture (Artur Jasinski): For any positive number x >= A179794(n), the distance d between the eleventh power of x and the square of any y (such that x <> k^2 and y <> k^11) can't be less than A179793(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.

d = 11; 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)]; k = n^d - m^2; 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, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx

A179795 Values y for records of the minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).

Original entry on oeis.org

45, 420, 19047, 44467, 92681, 316227, 2012353, 8016758, 14310835, 60583368, 91068707, 189812531, 488438379, 2741690265, 6023263700, 23751934582, 771834189385, 1734606819630, 8034176335637, 11511075516802, 22632960587688

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^11.
For x values see A179794.
For x values see A179795.
Conjecture (Artur Jasinski):
For any positive number x >= A179794(n), the distance d between the eleventh power of x and the square of any y (such that x <> k^2 and y <> k^11) can't be less than A179793(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.

d = 11; 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)]; k = n^d - m^2; 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, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy

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

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 15, 18, 19, 20, 23, 25, 26, 27, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 64, 67, 71, 72, 74, 76, 79, 81, 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 125, 126, 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153

Offset: 1

Zak Seidov, Apr 28 2005

nonn

A given a(n) can have multiple solutions with distinct (b,c), e.g., a=4 with b=2, c=2 (4 + 2^2 = 2^3) or with b=11, c=5 (4 + 11^2 = 5^3). (See also A181138.) Sequences A106266 and A106267 list the minimal values. - M. F. Hasler, Oct 04 2013
The cubes A000578 = (1, 8, 27, 64, ...) form a subsequence of this sequence, corresponding to b=0, a=c^3. If b=0 is excluded, these terms are not present, except for a few exceptions, a = 216, 343, 12167, ... (6^3 + 28^2 = 10^3, 7^3 + 13^2 = 8^3, 23^3 + 588^2 = 71^3, ...), cf. A038597 for the possible b-values. - M. F. Hasler, Oct 05 2013
This is the complement of A081121. The values do indeed correspond to solutions listed in Gebel's file. - M. F. Hasler, Oct 05 2013
B-file corrected following a remark by Alois P. Heinz, May 24 2019. A double-check would be appreciated in view of two values that were missing, for unknown reasons, in the earlier version of the b-file. - M. F. Hasler, Aug 10 2024

			a = 1,2,4,7,8,11,13,15,18,19,20,23,25,26,27,28,35,39,40,44,45,47,48,49,53, ...
b = 0,5,2,1,0, 4,70, 7, 3,18,14, 2,10, 1, 0, 6,36, 5,52, 9,96,13,4,524,26, ...
c = 1,3,2,2,2, 3,17, 4, 3, 7, 6, 3, 5, 3, 3, 4,11, 4,14, 5,21, 6, 4,65, 9, ...
Here are the values grouped together:
{{1, 0, 1}, {2, 5, 3}, {4, 2, 2}, {7, 1, 2}, {8, 0, 2}, {11, 4, 3}, {13, 70, 17}, {15, 7, 4}, {18, 3, 3}, {19, 18, 7}, {20, 14, 6}, {23, 2, 3}, {25, 10, 5}, {26, 1, 3}, {27, 0, 3}, {28, 6, 4}, {35, 36, 11}, {39, 5, 4}, {40, 52, 14}, {44, 9, 5}, {45, 96, 21}, {47, 13, 6}, {48, 4, 4}, {49, 524, 65}, {53, 26, 9}, {54, 17, 7}, {55, 3, 4}, {56, 76, 18}, {60, 2, 4}, {61, 8, 5}, {63, 1, 4}, {64, 0, 4}, {67, 110, 23}, {71, 21, 8}, ... }
a(2243) = 10000 = 25^3 - 75^2. - _M. F. Hasler_, Oct 05 2013, index corrected Aug 10 2024
a(136) = 366 = 11815^3 - 1284253^2 (has c/a(n) ~ 32.3); a(939) = 3607 = 244772^3 - 121099571^2 (has c/a(n) ~ 67.9); a(1090) = 4265 = 84521^3 - 24572364^2 (has c/a(n) ~ 19.8). - _M. F. Hasler_, Aug 10 2024

M. F. Hasler, Table of n, a(n) for n = 1..2500 (corrected and extended Aug 10 2024)
J. Gebel, Integer points on Mordell curves, negative k values [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A106266, A106267 for respective minimal values of b and c.
Cf. A023055: (Apparent) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2); A076438: n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1; A076440: n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd); A075772: Difference between n-th perfect power and the closest perfect power, etc.
Cf. A023055, A075772, A076438, A076440.
Cf. A054504, A081121, A081120; A179386 - A179388.

f[n_] := Block[{k = Floor[n^(1/3) + 1]}, While[k < 10^6 && !IntegerQ[ Sqrt[k^3 - n]], k++ ]; If[k == 10^6, 0, k]]; Select[ Range[ 154], f[ # ] != 0 &] (* Robert G. Wilson v, Apr 28 2005 *)

select( {is_A106265(a, L=99)=for(c=sqrtnint(a, 3), (a+9)*L, issquare(c^3-a, &b) && return(c))}, [1..199]) \\ The function is_A106265 returns 0 if n isn't a term, or else the c-value (A106267) which can't be zero if n is a term. The L-value can be used to increase the search limit but so far no instance is known that requires L>68. - M. F. Hasler, Aug 10 2024

a(n) = A106267(n)^3 - A106266(n)^2.

More terms from Robert G. Wilson v, Apr 28 2005

Definition corrected, solutions with b=0 added by M. F. Hasler, Sep 30 2013

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

A179406 Record minima of the positive distance d between the fifth power of a positive integer x and the square of an integer y such that d = x^5 - y^2 (x != k^2 and y != k^5).

Original entry on oeis.org

7, 19, 60, 341, 47776, 70378, 78846, 115775, 220898, 780231, 2242100, 11889984, 26914479, 50406928, 77146256, 80117392, 284679759, 595974650, 2071791247, 7825152599, 67944824923, 742629277177, 1709838230002, 2676465117663

Offset: 1

Artur Jasinski, Jul 13 2010

Distance d is equal to 0 when x = k^2 and y = k^5.
For x values see A179407.
For y values see A179408.
Conjecture (from Artur Jasinski): For any positive number x >= A179407(n), the distance d between the fifth power of x and the square of any y (such that x != k^2 and y != k^5) can't be less than A179406(n).

J. Blass, A Note on Diophantine Equation Y^2 + k = X^5, Math. Comp. 1976, Vol. 30, No. 135, pp. 638-640.
A. Bremner, On the Equation Y^2 = X^5 + k, Experimental Mathematics 2008 Vol. 17, No. 3, pp. 371-374.

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408.

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^5)^(1/2)]; k = n^5 - m^2; 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, 96001}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd

A181138 Least positive integer k such that n^2 + k is a cube.

Original entry on oeis.org

1, 7, 4, 18, 11, 2, 28, 15, 61, 44, 25, 4, 72, 47, 20, 118, 87, 54, 19, 151, 112, 71, 28, 200, 153, 104, 53, 271, 216, 159, 100, 39, 307, 242, 175, 106, 35, 359, 284, 207, 128, 47, 433, 348, 261, 172, 81, 535, 440, 343, 244, 143, 40, 566, 459, 350, 239, 126, 11

Offset: 0

Jason Earls, Oct 06 2010

a(n) = A070923(n) if n is not cube. Zak Seidov, Mar 26 2013

See A229618 for the range of this sequence. A179386 gives the range of b(n) = min{ a(m); m >= n }. The indices of jumps in this sequence are given in A179388 = { n | a(m)>a(n) for all m > n } = { 0, 5, 11, 181, 207, 225, 500, 524, 1586, ... }. - M. F. Hasler, Sep 26 2013

			a(11) = 4 because 11^2 + k is never a cube for k < 4, but 11^2 + 4 = 5^3. - _Bruno Berselli_, Jan 29 2013

Bruno Berselli, Table of n, a(n) for n = 0..1000 (Corrected Jan 19 2019)

Cf. A070923, A077116.

S:=[];
k:=1;
for n in [0..60] do
   while not IsPower(n^2+k,3) do
        k:=k+1;
   end while;
   Append(~S, k);
   k:=1;
end for;
S;  // Bruno Berselli, Jan 29 2013

Table[(1 + Floor[n^(2/3)])^3 - n^2, {n, 100}] (* Zak Seidov, Mar 26 2013 *)

A181138(n)=(sqrtnint(n^2,3)+1)^3-n^2 \\ Charles R Greathouse IV, Mar 26 2013

a(n) << n^(4/3). - Charles R Greathouse IV, Mar 26 2013

Extended to a(0)=1 by M. F. Hasler, Sep 26 2013

cp's OEIS Frontend

A179791 Values x for records of the minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).

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A179792 Values y for records of the minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).

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A179793 Records of minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).

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A179794 Values x for records of the minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).

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A179795 Values y for records of the minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).

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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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A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

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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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A179406 Record minima of the positive distance d between the fifth power of a positive integer x and the square of an integer y such that d = x^5 - y^2 (x != k^2 and y != k^5).

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