A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

112, 240, 368, 496, 624, 752, 880, 1008, 1136, 1264, 1392, 1520, 1648, 1776, 1904, 2032, 2160

Offset

1,1

Comments

The last term in this sequence is 2160. The reasons are as follows (let b, c, d, i, j, k, m, r, s, t, w, x, y and z be nonnegative integers).

For the Diophantine equation x^2 + y^2 + z^2 + w^7 = m:

(1) If m is not of the form 4^c * (8b + 7), then it follows from Legendre's three-square theorem that the equation has a solution with w = 0.

(2) 8b + 7 - 1^7 = 8b + 6. Then m = 8b + 7, the equation has a solution with w = 1.

(3) 4 * (8b + 7) - 1^7 = (8 * (4b + 3) + 3) = 8d + 3. Then m = 4 * (8b + 7), the equation has a solution with w = 1.

(4) For b >= 17, 16 * (8b + 7) - 3^7 = 8 * (16 * (b - 17) + 12) + 5 = 8i + 5. Then m = 16 * (8b + 7) and b >= 17, the equation has a solution with w = 3.

(5) 4^3 * (8b + 7) - 2^7 = 4^3 * (8b + 5). Then m = 4^3 * (8b + 7), the equation has a solution with w = 2. And 4^3 * (8b + 7) - 3^7 = 8 * (4^3 * (b - 4) + 38) + 5 = 8j + 5. Then m = 4^3 * (8b + 7) and b >= 4, the equation has a solution with w = 3.

(6) 4^4 * (8b + 7) - 2^7 = 4^3 * (8 * (4b + 3) + 3) = 4^3 * (8k + 3). 4^4 * (8b + 7) - 3^7 = 8 * (256b - 217) + 3 = 8r + 3. Then m = 4^4 * (8b + 7), the equation has a solution with w = 2 and when b > 0, the equation has a solution with w = 3.

(7) When c >= 5, 4^c * (8b + 7) - 2^7 = 4^3 * (8 * (b * 4^(c - 3) + 14 * 4^(c - 5) + 5) = 4^3 * (8s + 5). 4^c * (8b + 7) - 3^7 = 8 * (b * 4^(c - 3) + 14 * 4^(c - 3) - 273) + 3 = 8t + 3. Then n = 4^c * (8b + 7), the equation has solutions with w = 2 and 3.

In short, except for the 17 numbers in the sequence, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative 7th power.

Links

Table of n, a(n) for n=1..17.

Wikipedia, Legendre's three-square theorem

Formula

a(n) = 128n - 16 = 16 * A004771(n - 1), 1 <= n <= 17.

Mathematica

t1={};
Do[Do[If[x^2+y^2+z^2+w^7==n, AppendTo[t1, n]&&Break[]], {x, 0, n^(1/2)}, {y, x, (n-x^2)^(1/2)}, {z, y, (n-x^2-y^2)^(1/2)}, {w, 0, (n-x^2-y^2-z^2)^(1/7)}], {n, 0, 3000}];
t2={};
Do[If[FreeQ[t1, k]==True, AppendTo[t2, k]], {k, 0, 3000}];
t2

Crossrefs

Finite subsequence of A004215 and A296185.

Cf. A004771, A022552, A022557, A022561, A022566, A111151.

Sequence in context: A119684 A235887 A296579 * A296185 A211444 A270759

Adjacent sequences: A297967 A297968 A297969 * A297971 A297972 A297973

Keyword

nonn,fini,full

Author

XU Pingya, Jan 10 2018

Status

approved