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

%I A003294 M5446 #51 Feb 16 2025 08:32:27
%S A003294 353,651,706,1059,1302,1412,1765,1953,2118,2471,2487,2501,2604,2824,
%T A003294 2829,3177,3255,3530,3723,3883,3906,3973,4236,4267,4333,4449,4557,
%U A003294 4589,4942,4949,4974,5002,5208,5281,5295,5463,5491,5543,5648,5658
%N A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers.
%C A003294 Sequence gives solutions k to the Diophantine equation A^4 + B^4 + C^4 + D^4 = k^4.
%C A003294 Is this sequence the same as A096739? - _David Wasserman_, Nov 16 2007
%C A003294 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
%D A003294 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A003294 D. Wells, Curious and interesting numbers, Penguin Books, p. 139.
%H A003294 T. D. Noe, <a href="/A003294/b003294.txt">Table of n, a(n) for n = 1..4870</a> (using Wroblewski's results)
%H A003294 Simcha Brudno, <a href="https://doi.org/10.1017/S0305004100038470">A further example of A^4 + B^4 + C^4 + D^4 = E^4</a>, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.
%H A003294 Lee W. Jacobi and Daniel J. Madden, <a href="http://www.jstor.org/stable/27642446">On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4</a>, Amer. Math. Monthly 115 (2008) 220-236.
%H A003294 Kermit Rose and Simcha Brudno, <a href="https://doi.org/10.1090/S0025-5718-1973-0329184-2">More about four biquadrates equal one biquadrate</a>, Math. Comp., 27 (1973), 491-494.
%H A003294 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation4thPowers.html">Diophantine Equation 4th Powers</a>.
%H A003294 Jaroslaw Wroblewski, <a href="http://www.math.uni.wroc.pl/~jwr/eslp/414.txt">Exhaustive list of 1009 solutions to (4,1,4) below 222,000</a>.
%H A003294 <a href="/index/Di#Diophantine">Index to sequences related to diophantine equations</a> (4,1,4).
%e A003294 353^4 = 30^4 + 120^4 + 272^4 + 315^4.
%e A003294 651^4 = 240^4 + 340^4 + 430^4 + 599^4.
%e A003294 2487^4 = 435^4 + 710^4 + 1384^4 + 2420^4.
%e A003294 2501^4 = 1130^4 + 1190^4 + 1432^4 + 2365^4.
%e A003294 2829^4 = 850^4 + 1010^4 + 1546^4 + 2745^4.
%t A003294 fourthPowerSums = {};
%t A003294 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}];
%t A003294 Union @ fourthPowerSums (* _Vladimir Joseph Stephan Orlovsky_, May 19 2010 *)
%Y A003294 Cf. A039664 (subsequence, primitive), A096739.
%Y A003294 Cf. also A138760 (subsequence).
%K A003294 nonn,nice
%O A003294 1,1
%A A003294 _N. J. A. Sloane_
%E A003294 Corrected and extended by _Don Reble_, Jul 07 2007
%E A003294 More terms from _David Wasserman_, Nov 16 2007
%E A003294 Definition clarified by _Jonathan Sondow_, Apr 06 2008