A293283 - OEIS

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

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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.