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 A266152 Least positive integer y such that n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists. 14
 8, 1, 2, 17, 1, 3, 139, 19, 37, 1, 3, 9, 2, 7, 3, 1411, 1, 2, 2, 1, 5, 4, 387, 3, 1, 1, 4, 7, 9, 2, 35, 1, 33, 2, 6, 5, 1, 4, 3, 11, 1, 6, 2, 429, 2, 5, 11, 179, 73, 1, 15, 1, 4, 3, 11, 3, 5, 2, 3, 15, 5, 6, 7, 3, 1, 6, 4, 6337, 8, 16, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Conjecture: Any integer m can be written as x^4 - y^3 + z^2, where x, y and z are positive integers. This is slightly stronger than the conjecture in A266003. See also A266153 for a related sequence, and A266212 for a stronger conjecture. If n is a positive square, then a(n) = 1. - Altug Alkan, Dec 23 2015 LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. EXAMPLE a(0) = 8 since 0 = 4^4 - 8^3 + 16^2. a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2. a(15) = 1411 since 15 = 119^4 - 1411^3 + 51075^2. a(11019) = 71383 since 11019 = 4325^4 - 71383^3 + 3719409^2. MATHEMATICA SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]] Do[y=1; Label[bb]; Do[If[SQ[n+y^3-x^4], Print[n, " ", y]; Goto[aa]], {x, 1, (n+y^3)^(1/4)}]; y=y+1; Goto[bb]; Label[aa]; Continue, {n, 0, 70}] CROSSREFS Cf. A000290, A000578, A000583, A262827, A266003, A266004, A266153, A266212. Sequence in context: A156944 A227424 A248965 * A021127 A010155 A019607 Adjacent sequences:  A266149 A266150 A266151 * A266153 A266154 A266155 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 22 2015 STATUS approved

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Last modified October 23 12:54 EDT 2019. Contains 328345 sequences. (Running on oeis4.)