A096739 -id:A096739 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

			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.

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.

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).

Cf. A039664 (subsequence, primitive), A096739.

Cf. also A138760 (subsequence).

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 *)

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

A039664 Numbers k such that k^4 can be written as a sum of four positive 4th powers with no common factor.

Original entry on oeis.org

353, 651, 2487, 2501, 2829, 3723, 3973, 4267, 4333, 4449, 4949, 5281, 5463, 5491, 5543, 5729, 6167, 6609, 6801, 7101, 7209, 7339, 7703, 8373, 8433, 8493, 8517, 8577, 8637, 9137, 9243, 9431, 9519, 9639, 9797, 9877, 10419, 10939, 11681, 11757

Offset: 1

Felice Russo

nonn

T. D. Noe, Table of n, a(n) for n = 1..1003 (from Wroblewski, with duplicate n removed)
L. J. Lander, T. R. Parkin and J. L. Selfridge, A survey of equal sums of like powers, Math. Comp. vol. 21 no. 99, 1967, pp. 446-459, Table 1.
Eric Weisstein's World of Mathematics, Diophantine Equation - 4th powers
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000

Cf. A003294 (nonprimitive solutions allowed), A096739.

Edited by Don Reble, Jul 07 2007

Qualifier "positive" added to the name by Jianing Song, Jan 24 2020

A130022 Smallest natural number whose 4th power is the sum of n 4th powers of distinct natural numbers, or 0 if no such number exists.

Original entry on oeis.org

1, 0, 422481, 353, 15, 35, 25, 31, 37, 41, 35, 43, 39, 43, 47, 53, 55, 50, 50, 46, 48, 48, 50, 48, 50, 48, 52, 53, 55, 56, 54, 58, 58, 63, 65, 67, 70, 71, 73, 77, 81, 85, 87, 91, 93, 97, 101

Offset: 1

J. Lowell, Jun 15 2007

Cf. a(3) A003828, a(4) A096739, 3rd powers A130012, n-th powers A007666.

More terms from Martin Fuller, Jul 06 2007

A138760 Numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n.

Original entry on oeis.org

5491, 10982, 16473, 21964, 27455, 32946, 38437, 43928, 49419, 51361, 54910, 60401, 65892, 71383, 76874, 82365, 87856, 93347, 98838, 102722, 104329, 109820, 115311, 120802, 126293, 131784, 137275, 142766, 148257, 153748, 154083, 159239, 164730

Offset: 1

Jonathan Sondow, Mar 28 2008

nonn

Any multiple of a member is also a member. A member that is not a multiple of another member is called primitive. Using elliptic curves, Jacobi and Madden prove that there are infinitely many primitive members. According to them, the only primitive members less than 222,000 are 5491 (due to Brudno) and 51361 (due to Wroblewski).

			5491^4 = 5400^4 + (-2634)^4 + 1770^4 + 955^4 and 5491 = 5400 - 2634 + 1770 + 955, so 5491 is a member (Brudno).
51361^4 = 48150^4 + (-31764)^4 + 27385^4 + 7590^4 and 51361 = 48150 - 31764 + 27385 + 7590, so 51361 is a member (Wroblewski).
1347505009^4 = 1338058950^4 + (-89913570)^4 + 504106884^4 + (-404747255)^4, and 1347505009 = 1338058950 - 89913570 + 504106884 - 404747255, so 1347505009 is a member (Jacobi-Madden).

Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.
Noam Elkies, On A^4 + B^4 + C^4 = D^4, Math. Comp. 51 (1988) 825-835.
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.
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4
Eric Weisstein's MathWorld, Diophantine equation - 4th powers
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000

Cf. A003294, A003828, A096739.

n^4 = a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 with abcd =/= 0.

A239247 Numbers n such that n^4 can be written as a sum of five distinct positive 4th powers.

Original entry on oeis.org

15, 30, 35, 45, 55, 60, 65, 70, 75, 85, 89, 90, 95, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 165, 170, 175, 178, 180, 185, 190, 195, 205, 210, 215, 220, 225, 230, 233, 235, 240, 245, 250, 255, 260, 265, 267, 270, 275, 280, 285, 290, 295, 300

Offset: 1

Michel Marcus, Mar 13 2014

nonn

Every multiple of a term is a term.

			15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4.
35^4 = 4^4 + 21^4 + 22^4 + 26^4 + 28^4.
55^4 = 2^4 + 13^4 + 16^4 + 44^4 + 48^4.
65^4 = 1^4 + 8^4 + 12^4 + 32^4 + 64^4.
85^4 = 2^4 + 13^4 + 32^4 + 34^4 + 84^4.
89^4 = 10^4 + 35^4 + 52^4 + 60^4 + 80^4.
95^4 = 6^4 + 48^4 + 66^4 + 67^4 + 78^4.
115^4 = 4^4 + 31^4 + 48^4 + 58^4 + 112^4.
125^4 = 8^4 + 11^4 + 26^4 + 84^4 + 118^4.
145^4 = 2^4 + 23^4 + 46^4 + 52^4 + 144^4.
155^4 = 6^4 + 39^4 + 88^4 + 96^4 + 144^4.
185^4 = 2^4 + 38^4 + 62^4 + 87^4 + 182^4.
205^4 = 4^4 + 133^4 + 142^4 + 146^4 + 156^4.
215^4 = 4^4 + 26^4 + 127^4 + 174^4 + 176^4.
233^4 = 40^4 + 65^4 + 94^4 + 150^4 + 220^4.
235^4 = 9^4 + 52^4 + 148^4 + 184^4 + 194^4.

Lars Blomberg, Table of n, a(n) for n = 1..7560 (terms < 10000)
Lars Blomberg, Primitive solutions for terms < 10000. Only the first solution found for each term is included.
Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.

Cf. A130022, A003828 (three 4th powers), A096739 (four 4th powers).

isok(n) = {ret = 0; for (x=1, sqrtnint(n^4\5, 4), for (y=x+1, sqrtnint((n^4 - x^4)\4, 4), for (z=y+1, sqrtnint((n^4 - x^4 - y^4)\3, 4), for (t=z+1, sqrtnint((n^4 - x^4 - y^4 - z^4)\2, 4), for (u=t+1, sqrtnint((n^4 - x^4 - y^4 - z^4 - t^4), 4), if (x^4+y^4+z^4+t^4+u^4 == n^4, print(n, ": ", x, ", ", y, ", ",z ,", ",t, ", ",u); ret = 1;);););););); return (ret);}

a(1) = A130022(4).

Missing terms 15 and its multiples found by Alois P. Heinz, Mar 14 2014
More examples from Michel Marcus, Mar 18 2014
More terms from Lars Blomberg, Apr 05 2014

A134820 Primes in the sequence A003294 of certain fourth powers bases.

Original entry on oeis.org

353, 5281, 7703, 9137, 9431, 10939, 11681, 14029, 14489, 17519, 17881, 18077, 18701, 19483, 20719, 21013, 22247, 22961, 24953, 25589, 25913, 26821, 26987, 27893, 28297, 34327, 37273, 39671, 40031, 40129, 42209, 42359, 43397, 43781

Offset: 1

R. J. Mathar, Jan 28 2008

nonn

J. Wroblewski in Puzzle47 of primepuzzles.net.

Cf. A039664, A003294, A096739.

A003294 INTERSECT A000040

cp's OEIS Frontend

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

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A039664 Numbers k such that k^4 can be written as a sum of four positive 4th powers with no common factor.

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A130022 Smallest natural number whose 4th power is the sum of n 4th powers of distinct natural numbers, or 0 if no such number exists.

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A138760 Numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n.

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A239247 Numbers n such that n^4 can be written as a sum of five distinct positive 4th powers.

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A134820 Primes in the sequence A003294 of certain fourth powers bases.

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