A113505 Numbers not the sum of at most three perfect powers (A001597).

7, 15, 23, 87, 111, 119, 167, 335, 1391, 1455, 1607, 1679, 1991, 25887, 26375

Offset: 1

Author

R. P. van der Hilst (R.P.vanderHilst(AT)students.uu.nl), Jan 12 2006

Keywords

• nonn
• more

Comments

Cannot be written in the form a^x + b^y + c^z with a, b, c >= 0 and x, y, z > 1.

a(16), if it exists, is larger than 10^8. - Giovanni Resta, May 07 2017

From Brian Trial, Jun 07 2025: (Start)

Per Legendre's three-square theorem (A004215) only integers of the form 4^i(8j+7) are eligible.

Every integer > 5042631 (= 1424^2 + 734*2 + 19^5) and < 10^9 can be expressed as either a^2 + b^2 + c^2 or a^2 + b^2 + c^3, a,b,c >= 0 so a(16) >= 10^9. (End)

Links

• Eric Weisstein's World of Mathematics, Perfect Power
• Eric Weisstein's World of Mathematics, Powerful Number.

Crossrefs

A056828 is a subset, A001694, A274459.

Programs

Mathematica

lmt = 40000; s = Union@ Join[{0, 1}, Flatten@ Table[n^i, {n, 2, Sqrt@ lmt}, {i, 2, Log[n, lmt]}]]; t = Select[ Union[Plus @@@ Tuples[s, 3]], # < lmt + 1 &]; Complement[Range@ lmt, t] (* Robert G. Wilson v *)

Extensions

Edited by Robert G. Wilson v, May 01 2006