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

Author

Zhi-Wei Sun, Jan 02 2016

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

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

Programs

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}]