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A214891 Numbers that are not the sum of two squares and two fourth powers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From XU Pingya, Feb 07 2018: (Start)

When n is a term, 16n is also. This can been 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)

LINKS

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

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.

Sequence in context: A138975 A168439 A198949 * 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

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Last modified January 22 16:29 EST 2020. Contains 331152 sequences. (Running on oeis4.)