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A227855 Numbers of the form x^4 + 4*y^4. 3
0, 1, 4, 5, 16, 20, 64, 65, 80, 81, 85, 145, 256, 260, 320, 324, 325, 340, 405, 580, 625, 629, 689, 949, 1024, 1025, 1040, 1105, 1280, 1296, 1300, 1360, 1620, 1649, 2320, 2401, 2405, 2465, 2500, 2501, 2516, 2581, 2725, 2756, 3125, 3425, 3796, 4096, 4100, 4160 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Since 4 is even, either x or y or both may be negative integers, because their fourth powers will then be positive.

The only prime term in this sequence is 5; this can be proved using Sophie Germain's identity.

REFERENCES

Titu Andreescu and Rǎzvan Gelca, Mathematical Olympiad Challenges, New York, Birkhäuser (2009), p. 48.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Graeme Taylor, Identity of Sophie Germain, April 1, 2006.

FORMULA

x^4 + 4y^4 = (x^2 - 2xy + 2y^2)(x^2 + 2xy + 2y^2). This is Sophie Germain's identity.

EXAMPLE

80 = 2^4 + 4 * 2^4.

81 = 3^4 + 4 * 0^4.

85 = 3^4 + 4 * 1^4.

MATHEMATICA

nn = 10; Union[Select[Flatten[Table[x^4 + 4*y^4, {x, 0, nn}, {y, 0, nn}]], # <= nn^4 &]] (* T. D. Noe, Nov 08 2013 *)

PROG

(PARI) list(lim)=my(v=List(), t); for(y=0, sqrtnint(lim\4, 4), for(x=0, sqrtnint(lim\1-(t=4*y^4), 4), listput(v, t+x^4))); Set(v) \\ Charles R Greathouse IV, Nov 07 2013

CROSSREFS

Subsequences include A000583, A141046 and A001589.

Sequence in context: A137527 A024854 A025617 * A078581 A092809 A250254

Adjacent sequences:  A227852 A227853 A227854 * A227856 A227857 A227858

KEYWORD

nonn,easy

AUTHOR

Alonso del Arte, Oct 31 2013

STATUS

approved

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Last modified February 20 16:25 EST 2018. Contains 299380 sequences. (Running on oeis4.)