VOOZH about

URL: https://oeis.org/A266651

⇱ A266651 - OEIS

login
A266651
Nonnegative integers x such that x^3 + 6^3 is a sum of two squares.
2
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
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}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 02 2016
STATUS
approved