A117594 Numbers whose fifth powers are closer to cubic numbers than square numbers.

199, 1354, 4995, 7320, 7994, 12634, 44217, 91116, 177682, 394826, 458908, 462763, 512012, 1706886, 1738064, 1801677, 1880465, 2523441, 5691648, 6714911, 8383950, 8403388, 11100341, 14706104, 14706146, 15460136, 16337238, 18898872, 21194961

Offset: 1

Author

Ed Pegg Jr, Apr 05 2006

Keywords

• nonn

Comments

Numbers which are cubes themselves are excluded as trivial.

It appears that this sequence is infinite. For seventh powers < 10^49, only 2^7 and 3^7 are closer to cubes than squares. Note that 1/2+1/3+1/5>1, but 1/2+1/3+1/7<1. Do these inequalities determine whether there are an infinite or finite number of solutions? Mazur discusses how the ABC conjecture applies to perfect power problems. - T. D. Noe, Apr 07 2006

Examples

			The distance of 199^5 to the nearest cube is 49688. To the nearest square is 165882.
		

Links

• B. Mazur, Questions about Number
• Eric Weisstein's World of Mathematics, MathWorld: Perfect Power

Crossrefs

Cf. A117934 (perfect powers that are close).

Programs

Mathematica

nMax=10^5; lst={}; Do[n5=n^5; n3=Round[n5^(1/3)]^3; n2=Round[n5^(1/2)]^2; If[0 < Abs[n5-n3] < Abs[n5-n2], AppendTo[lst,n]], {n,nMax}]; lst (* T. D. Noe, Apr 07 2006 *)

Extensions

More terms from T. D. Noe and Hans Havermann, Apr 08 2006