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A003294
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Numbers n such that n^4 can be written as a sum of four positive 4th powers.
(Formerly M5446)
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7
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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
(list;
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listen;
history;
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OFFSET
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1,1
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COMMENTS
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Sequence gives solutions n to the Diophantine equation A^4 + B^4 + C^4 + D^4 = n^4.
Is this sequence the same as A096739? - David Wasserman, Nov 16 2007
A138760 (numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n) is a subsequence. - Jonathan Sondow, Apr 06 2008
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REFERENCES
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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.
K. Rose and S. Brudno, More about four biquadrates equal one biquadrate, Math. Comp., 27 (1973), 491-494.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..4870 (using Wroblewski's results)
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4
Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000
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EXAMPLE
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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.
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MATHEMATICA
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lst={}; 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[lst, e]], {d, c+1, 9000}], {c, b+1, 6000}], {b, a+1, 5000}], {a, 30, 3000}]; Union@lst [From Vladimir Joseph Stephan Orlovsky, May 19 2010]
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CROSSREFS
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Cf. A039664, A096739.
Cf. also A138760.
Sequence in context: A177678 A058375 A059635 * A096739 A039664 A054825
Adjacent sequences: A003291 A003292 A003293 * A003295 A003296 A003297
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Corrected and extended by Don Reble (djr(AT)nk.ca), Jul 07 2007
More terms from David Wasserman, Nov 16 2007
Definition clarified by Jonathan Sondow, Apr 06 2008
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STATUS
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approved
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