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%I #31 Jul 07 2023 21:22:36
%S 1,5,7,17,42,71,78,97,215,307,383,433,545,804,1351,1727,1777,1849,
%T 2039,2316,2387,2395,2899,2981,3331,4446,5441,7057,7189,7321,8023,
%U 8135,9847,9959,11052,13175,13679,17514,18143,18817,18929,19110,20593,22247,23417,23783,24407,24648
%N Numbers k such that k^4 = x^3 + y^2 has an integer solution.
%H Darío Alpern, <a href="https://www.alpertron.com.ar/SPOW432.HTM">List of first 1602 solutions to a^4 + b^3 = c^2 for increasing values of a</a>
%e 433 is in the sequence since we have 433^4 + 26462^3 = 4308693^2.
%t 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 *)
%K nice,nonn
%O 1,2
%A _Lekraj Beedassy_, May 30 2002
%E Example fixed by _Jean-François Alcover_, Jan 23 2013
%E Terms 7, 1351, 2899, 2981, 18929, 24648 inserted by _Jean-François Alcover_, Jan 23 2013
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