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A096741 Numbers n such that n^4 = x^3 + y^2 has a solution in integers. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

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

Dario 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 * A106955 A030785 A019404

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

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Last modified February 21 10:49 EST 2019. Contains 320372 sequences. (Running on oeis4.)