login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A214891 Numbers that are not the sum of two squares and two fourth powers. 4

%I #50 Apr 21 2023 12:47:38

%S 23,44,71,79,184,368,519,599,704,1136,1264,2944,4024,5888,8304,9584,

%T 11264,18176,20224,47104,64384,94208,132864,153344,180224,290816,

%U 323584,753664,1030144,1507328,2125824,2453504,2883584,4653056,5177344,12058624,16482304

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

%C From _XU Pingya_, Feb 07 2018: (Start)

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

%C (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).

%C (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.

%C (End)

%C 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

%H Donovan Johnson, <a href="/A214891/b214891.txt">Table of n, a(n) for n = 1..52</a> (terms <= 4*10^9)

%H Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/414791">On w^4+x^4+y^2+z^2 over a number field</a>, Question 414791 at MathOverflow, Jan. 27, 2022.

%o (PARI)

%o N=10^6; x='x+O('x^N);

%o S(e)=sum(j=0, ceil(N^(1/e)), x^(j^e));

%o v=Vec( S(4)^2 * S(2)^2 );

%o for(n=1,#v,if(!v[n],print1(n-1,", ")));

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

%K nonn

%O 1,1

%A _Joerg Arndt_, Jul 29 2012

%E a(29)-a(37) from _Donovan Johnson_, Jul 29 2012

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)