A274459 -id:A274459 - cp's OEIS frontend

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

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

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)

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

A056828 is a subset, A001694, A274459.

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 *)

Edited by Robert G. Wilson v, May 01 2006

A354761 Least number of squares and cubes that add up to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 1, 2, 3, 3, 2, 3

Offset: 1

Peter Schorn, Jun 06 2022

nonn

a(n) <= 4 since any number can be written as a sum of 4 squares (Lagrange's theorem).

Sequence first differs from A063274, A225926 and A274459 at n = 32 since 32 is a powerful number, a prime power and a perfect power but neither a square nor a cube.

			a(1) = 1, a(4) = 1 (4 = 2^2), a(7) = 4 (7 = 2^2 + 1^2 + 1^2 + 1^2), a(8) = 1 (8 = 2^3), a(12) = 2 (12 = 2^3 + 2^2), a(17) = 2 (17 = 4^2 + 1^2), a(32) = 2 (32 = 4^2 + 4^2).

Cf. A002760, A063274, A225926, A274459, A002376, A002828, A000290, A000578.

lista(n) = {my(v = vector(n)); for(j = 2, 3, for(i = 2, sqrtnint(n, j), v[i^j] = 1)); v[1]=1; v[2]=2; for(i=3, #v, if(v[i]==0, v[i] = vecmin(vector(i\2, k, v[k] + v[i-k])))); v}

cp's OEIS Frontend

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

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