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 A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n. 3
 6, 9, 18, 40, 42, 68, 75, 90, 99, 105, 122, 126, 130, 174, 192, 196, 225, 251, 257, 288, 315, 325, 330, 350, 405, 490, 499, 504, 516, 528, 546, 550, 576, 614, 651, 665, 684, 726, 735, 744, 849, 882, 900, 920, 936, 974, 1025, 1032, 1036, 1107, 1140, 1183, 1200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For n > 0, k = (n + 1)(2n + 1)^2 is a term in this sequence, because k^2 = (n * (2n + 1)^2)^2 + (2n + 1)^5. Examples: 18, 75, 196, 405, 726, 1183. When z^2 = x^2 + y^2 (i.e., z = A009003(n)), (z * y^4)^2 = (x * y^4)^2 + (y^2)^5. Thus z * y^4 is a term in this sequence. For example, 1200. More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 1) * z^(5k - 5) is in this sequence. When z^2 = x^2 + y^3 (i.e., z = A070745(n)), (z * y)^2 = (x * y)^2 + y^5. Thus z * y is in this sequence. E.g. 6, 18, 40, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 4) * z^(5k - 4) is in this sequence. When z^2 = x^2 + y^4 (i.e., z = A271576(n)), (z * y^3)^2 = (x * y^3)^2 + (y^2)^5. Thus z * y^3 is also in this sequence. E.g. 40, 405, 1107, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 2) * z^(5k - 4) is in this sequence. LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 EXAMPLE 6^2 = 2^2 + 2^5. 9^2 = 7^2 + 2^5. MATHEMATICA c[n_]: = Count[n^2 - Range[(n^2 - 1)^(1/5)]^5, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range, c] PROG (PARI) isok(n) = for (k=1, n-1, if (ispower(n^2-k^2, 5), return (1)); ); return (0); \\ Michel Marcus, Oct 06 2017 CROSSREFS Cf. A009003, A070067, A070745, A228946, A271576, A274035. Sequence in context: A086955 A270794 A297623 * A078415 A265099 A023041 Adjacent sequences: A293280 A293281 A293282 * A293284 A293285 A293286 KEYWORD nonn AUTHOR XU Pingya, Oct 04 2017 STATUS approved

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Last modified September 22 11:22 EDT 2023. Contains 365521 sequences. (Running on oeis4.)