A096741 Numbers k such that k^4 = x^3 + y^2 has an integer solution.

1, 5, 7, 17, 42, 71, 78, 97, 215, 307, 383, 433, 545, 804, 1351, 1727, 1777, 1849, 2039, 2316, 2387, 2395, 2899, 2981, 3331, 4446, 5441, 7057, 7189, 7321, 8023, 8135, 9847, 9959, 11052, 13175, 13679, 17514, 18143, 18817, 18929, 19110, 20593, 22247, 23417, 23783, 24407, 24648

Offset

1,2

Links

Table of n, a(n) for n=1..48.

Darío Alpern, List of first 1602 solutions to a^4 + b^3 = c^2 for increasing values of a

Example

433 is in the sequence since we have 433^4 + 26462^3 = 4308693^2.

Mathematica

a[1][s_, t_] := 6*s^5*t + 72*s*t^5; a[2][s_, t_] := 18*s^5*t + 24*s*t^5; a[3][s_, t_] := -s^6 + 40*s^3*t^3 + 32*t^6; a[4][s_, t_] := -s^6 - 15*s^4*t^2 + 45*s^2*t^4 + 27*t^6; a[5][s_, t_] := -s^6 + 6*s^5*t + 15*s^4*t^2 + 20*s^3*t^3 - 15*s^2*t^4 + 30*s*t^5 + 17*t^6; a[6][s_, t_] := -5*s^6 - 6*s^5*t + 15*s^4*t^2 + 60*s^3*t^3 + 45*s^2*t^4 + 18*s*t^5 + 9*t^6; a[7][s_, t_] := -7*s^6 - 6*s^5*t + 30*s^4*t^2 + 80*s^3*t^3 + 60*s^2*t^4 + 24*s*t^5 + 8*t^6; b[1][s_, t_] := s^8 - 168*s^4*t^4 + 144*t^8; b[2][s_, t_] := 9*s^8 - 168*s^4*t^4 + 16*t^8; b[3][s_, t_] := 8*s^7*t + 112*s^4*t^4 - 256*s*t^7; b[4][s_, t_] := -s^8 + 28*s^6*t^2 + 42*s^4*t^4 + 252*s^2*t^6 - 81*t^8; b[5][s_, t_] := 2*s^8 + 8*s^7*t + 56*s^5*t^3 - 28*s^4*t^4 - 168*s^3*t^5 - 112*s^2*t^6 - 88*s*t^7 + 42*t^8; b[6][s_, t_] := 6*s^8 + 56*s^7*t + 112*s^6*t^2 + 168*s^5*t^3 + 252*s^4*t^4 + 168*s^3*t^5 - 72*s*t^7 - 18*t^8; b[7][s_, t_] := 15*s^8 + 104*s^7*t + 224*s^6*t^2 + 336*s^5*t^3 + 392*s^4*t^4 + 224*s^3*t^5 - 64*s*t^7 - 16*t^8; tab = Table[{a[k][s, t] // Abs, -b[k][s, t]}, {k, 1, 7}, {s, -5, 8}, {t, 0, 5}] // Flatten[#, 2] & // Select[#, 0 < #[[1]] < 25000 &] & // Union; r[{n_, x_}] := (rn = Reduce[y > 0 && GCD[n, x, y] == 1 && n^4 == x^3 + y^2, y, Integers]; If[rn =!= False, {rn, {n, x, y} /. ToRules[rn]}, {False, 0, 0}]); Select[tab, r[#][[1]] =!= False &] [[All, 1]] // Rest (* Jean-François Alcover, Jan 23 2013, after Darío Alpern *)

Crossrefs

Sequence in context: A163570 A113282 A323200 * A355188 A106955 A030785

Adjacent sequences: A096738 A096739 A096740 * A096742 A096743 A096744

Keyword

nice,nonn

Author

Lekraj Beedassy, May 30 2002

Extensions

Example fixed by Jean-François Alcover, Jan 23 2013

Terms 7, 1351, 2899, 2981, 18929, 24648 inserted by Jean-François Alcover, Jan 23 2013

Status

approved