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A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers.
(Formerly M5446)
18
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; graph; refs; listen; history; text; internal format)
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.
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
KEYWORD
nonn,nice
AUTHOR
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

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)