A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers. (Formerly M5446)

353, 651, 706, 1059, 1302, 1412, 1765, 1953, 2118, 2471, 2487, 2501, 2604, 2824, 2829, 3177, 3255, 3530, 3723, 3883, 3906, 3973, 4236, 4267, 4333, 4449, 4557, 4589, 4942, 4949, 4974, 5002, 5208, 5281, 5295, 5463, 5491, 5543, 5648, 5658

Offset

1,1

Comments

Sequence gives solutions k to the Diophantine equation A^4 + B^4 + C^4 + D^4 = k^4.

Is this sequence the same as A096739? - David Wasserman, Nov 16 2007

A138760 (numbers k such that k^4 is a sum of 4th powers of four nonzero integers whose sum is k) is a subsequence. - Jonathan Sondow, Apr 06 2008

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Wells, Curious and interesting numbers, Penguin Books, p. 139.

Links

T. D. Noe, Table of n, a(n) for n = 1..4870 (using Wroblewski's results)

Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.

Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220-236.

Kermit Rose and Simcha Brudno, More about four biquadrates equal one biquadrate, Math. Comp., 27 (1973), 491-494.

Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.

Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000.

Index to sequences related to diophantine equations (4,1,4).

Example

353^4 = 30^4 + 120^4 + 272^4 + 315^4.
651^4 = 240^4 + 340^4 + 430^4 + 599^4.
2487^4 = 435^4 + 710^4 + 1384^4 + 2420^4.
2501^4 = 1130^4 + 1190^4 + 1432^4 + 2365^4.
2829^4 = 850^4 + 1010^4 + 1546^4 + 2745^4.

Mathematica

fourthPowerSums = {};
Do[a4 = a^4; Do[b4 = b^4; Do[c4 = c^4; Do[d4 = d^4; e4 = a4 + b4 + c4 + d4; e = Sqrt[Sqrt[e4]]; If[IntegerQ[e], AppendTo[fourthPowerSums, e]], {d, c + 1, 9000}], {c, b + 1, 6000}], {b, a + 1, 5000}], {a, 30, 3000}];
Union @ fourthPowerSums (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)

Crossrefs

Cf. A039664 (subsequence, primitive), A096739.

Cf. also A138760 (subsequence).

Sequence in context: A177678 A058375 A059635 * A096739 A039664 A054825

Adjacent sequences: A003291 A003292 A003293 * A003295 A003296 A003297

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Corrected and extended by Don Reble, Jul 07 2007

More terms from David Wasserman, Nov 16 2007

Definition clarified by Jonathan Sondow, Apr 06 2008

Status

approved