|
%I M5446
%S 353,651,706,1059,1302,1412,1765,1953,2118,2471,2487,2501,2604,2824,
%T 2829,3177,3255,3530,3723,3883,3906,3973,4236,4267,4333,4449,4557,
%U 4589,4942,4949,4974,5002,5208,5281,5295,5463,5491,5543,5648,5658
%N Numbers n such that n^4 can be written as a sum of four positive 4th powers.
%C Sequence gives solutions n to the Diophantine equation A^4 + B^4 + C^4 + D^4 = n^4.
%C Is this sequence the same as A096739? - _David Wasserman_, Nov 16 2007
%C 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
%D Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.
%D 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.
%D K. Rose and S. Brudno, More about four biquadrates equal one biquadrate, Math. Comp., 27 (1973), 491-494.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D D. Wells, Curious and interesting numbers, Penguin Books, p. 139.
%H T. D. Noe, <a href="/A003294/b003294.txt">Table of n, a(n) for n=1..4870</a> (using Wroblewski's results)
%H Lee W. Jacobi and Daniel J. Madden, <a href="http://www.maa.org/pubs/monthly_mar08_toc.html">On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation4thPowers.html">Diophantine Equation 4th Powers.</a>
%H 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>
%e 353^4 = 30^4 + 120^4 + 272^4 + 315^4.
%e 651^4 = 240^4 + 340^4 + 430^4 + 599^4.
%e 2487^4 = 435^4 + 710^4 + 1384^4 + 2420^4.
%e 2501^4 = 1130^4 + 1190^4 + 1432^4 + 2365^4.
%e 2829^4 = 850^4 + 1010^4 + 1546^4 + 2745^4.
%t 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]
%Y Cf. A039664, A096739.
%Y Cf. also A138760.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_.
%E Corrected and extended by Don Reble (djr(AT)nk.ca), Jul 07 2007
%E More terms from _David Wasserman_, Nov 16 2007
%E Definition clarified by _Jonathan Sondow_, Apr 06 2008
|
