A293283 - OEIS

%I #22 Oct 08 2017 13:15:01

%S 6,9,18,40,42,68,75,90,99,105,122,126,130,174,192,196,225,251,257,288,

%T 315,325,330,350,405,490,499,504,516,528,546,550,576,614,651,665,684,

%U 726,735,744,849,882,900,920,936,974,1025,1032,1036,1107,1140,1183,1200

%N Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

%C 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.

%C 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.

%C 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.

%C 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.

%H Chai Wah Wu, <a href="/A293283/b293283.txt">Table of n, a(n) for n = 1..10000</a>

%e 6^2 = 2^2 + 2^5.

%e 9^2 = 7^2 + 2^5.

%t c[n_]: = Count[n^2 - Range[(n^2 - 1)^(1/5)]^5, _?(IntegerQ[Sqrt[#]] &)] > 0;

%t Select[Range[1200], c]

%o (PARI) isok(n) = for (k=1, n-1, if (ispower(n^2-k^2, 5), return (1));); return (0); \\ _Michel Marcus_, Oct 06 2017

%Y Cf. A009003, A070067, A070745, A228946, A271576, A274035.

%K nonn

%O 1,1

%A _XU Pingya_, Oct 04 2017