A266651 Nonnegative integers x such that x^3 + 6^3 is a sum of two squares.

14, 21, 62, 190, 206, 210, 237, 286, 334, 350, 382, 398, 426, 430, 446, 453, 574, 622, 670, 734, 766, 777, 782, 878, 958, 974, 1102, 1294, 1317, 1342, 1438, 1486, 1678, 1694, 1722, 1749, 1774, 1790, 1938, 1965, 1966, 2014, 2030, 2110, 2126, 2154, 2222, 2254, 2270, 2289, 2302, 2397, 2414, 2446, 2558, 2638, 2686, 2721, 2734, 2750

Offset

1,1

Comments

Conjecture: For any integer x with gcd(x,6) = 1, the number x^3 + 6^3 is never a sum of two squares.

We have verified this for x up to 5*10^6.

Note also that 6^3 + (-2)^3 = 8^2 + 12^2.

Hao Pan at Nanjing Univ. confirmed the conjecture on Jan. 3, 2016. - Zhi-Wei Sun, Jan 06 2016

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(1) = 14 since 14^3 + 6^3 = 16^2 + 52^2.
a(7) = 237 since 237^3 + 6^3 = 162^2 + 3645^2.

Mathematica

f[n_]:=f[n]=FactorInteger[n]
Le[n_]:=Le[n]=Length[f[n]]
n=0; Do[Do[If[Mod[Part[Part[f[x^3+6^3], i], 1], 4]==3&&Mod[Part[Part[f[x^3+6^3], i], 2], 2]==1, Goto[aa]], {i, 1, Le[216+x^3]}]; n=n+1; Print[n, " ", x]; Label[aa]; Continue, {x, 0, 2750}]

Crossrefs

Cf. A000290, A000578, A001481, A266152, A266230, A266231, A266277, A266363, A266364, A266548.

Sequence in context: A073250 A335760 A159453 * A351689 A166628 A199250

Adjacent sequences: A266648 A266649 A266650 * A266652 A266653 A266654

Keyword

nonn

Author

Zhi-Wei Sun, Jan 02 2016

Status

approved