A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

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

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[1200], 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