A214891 Numbers that are not the sum of two squares and two fourth powers.

23, 44, 71, 79, 184, 368, 519, 599, 704, 1136, 1264, 2944, 4024, 5888, 8304, 9584, 11264, 18176, 20224, 47104, 64384, 94208, 132864, 153344, 180224, 290816, 323584, 753664, 1030144, 1507328, 2125824, 2453504, 2883584, 4653056, 5177344, 12058624, 16482304

Offset

1,1

Comments

From XU Pingya, Feb 07 2018: (Start)

When n is a term, 16n is also. This can be proved as follows:

(1) If w is odd, then 16n - w^4 == 7 (mod 8), and it follows from Legendre's three-square theorem that the equation x^2 + y^2 + z^4 + w^4 = 16n has no solution (it is the same when x, y or z are odd numbers).

(2) If x, y, z and w are even numbers (x = 2a, y = 2b, z = 2c, w = 2d) such that x^2 + y^2 + z^4 + w^4 = 16n, then a^2 + b^2 = 4(n - c^4 - d^4). So there are integers u and v satisfying u^2 + v^2 = n - c^4 - d^4. i.e. u^2 + v^2 + c^4 + d^4 = n, which is a contradiction.

(End)

Conjecture: The set {a(n): n > 0} coincides with {16^k*m: k = 0, 1, 2, ... and m = 23, 44, 71, 79, 184, 519, 599, 4024}. - Zhi-Wei Sun, Jan 27 2022

Links

Donovan Johnson, Table of n, a(n) for n = 1..52 (terms <= 4*10^9)

Zhi-Wei Sun, On w^4+x^4+y^2+z^2 over a number field, Question 414791 at MathOverflow, Jan. 27, 2022.

Prog

(PARI)
N=10^6;  x='x+O('x^N);
S(e)=sum(j=0, ceil(N^(1/e)), x^(j^e));
v=Vec( S(4)^2 * S(2)^2 );
for(n=1, #v, if(!v[n], print1(n-1, ", ")));

Crossrefs

Cf. A001481, A004999, A022549, A346643, A347865, A350857, A350860.

Sequence in context: A332399 A198949 A334113 * A003859 A058545 A161709

Adjacent sequences: A214888 A214889 A214890 * A214892 A214893 A214894

Keyword

nonn

Author

Joerg Arndt, Jul 29 2012

Extensions

a(29)-a(37) from Donovan Johnson, Jul 29 2012

Status

approved