A261029 -id:A261029 - cp's OEIS frontend

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

72, 94, 107, 144, 365, 415, 427, 435, 480, 503, 530, 553, 575, 650, 700, 703, 716, 729, 744, 764, 804, 848, 851, 875, 923, 941, 975, 1004, 1006, 1040, 1044, 1235, 1257, 1313, 1327, 1329, 1369, 1392, 1457, 1469, 1504, 1528, 1537, 1575, 1583, 1588, 1596, 1623, 1653, 1685, 1686

Offset: 1

David W. Wilson, Aug 31 2001

nonn

Primitive means a solution for k has gcd(a,b,c,d,e) = 1. [Corrected by Jianing Song, Jan 24 2020]

Nontrivial means at least two of a,b,c,d,e are nonzero. - Jianing Song, Jan 24 2020

			   72^5 = 19^5 + 43^5 + 46^5 + 47^5 +  67^5;
   94^5 = 21^5 + 23^5 + 37^5 + 79^5 +  84^5;
  107^5 =  7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Jianing Song, Table of n, a(n) for n = 1..400 (All terms from James Waldby link)
L. J. Lander, T. R. Parkin and J. L. Selfridge, A Survey of Equal Sums of Like Powers, Mathematics of Computation, Vol. 21, No. 99 (Jul., 1967), pp. 446-459. See Table III p. 449.
James Waldby, A Table of Fifth Powers equal to a Fifth Power
Index to sequences related to Diophantine equations (5,1,5)

Cf. A063922.
For cubes: A003072, A023041, A261029.
For fourth powers: A003828, A175610, A039664, A003294.

144 and 1006 inserted and name simplified by Jianing Song, Jan 24 2020

More terms from Jinyuan Wang, Jan 24 2020

A330013 a(n) is the number of solutions with nonnegative (x,y,z) to the cubic Diophantine equation x^3+y^3+z^3 - 3xy*z = n.

Original entry on oeis.org

3, 3, 0, 3, 3, 0, 3, 6, 6, 3, 3, 0, 3, 3, 0, 6, 3, 6, 3, 6, 0, 3, 3, 0, 3, 3, 9, 12, 3, 0, 3, 6, 0, 3, 9, 6, 3, 3, 0, 6, 3, 0, 3, 6, 6, 3, 3, 0, 9, 3, 0, 6, 3, 12, 3, 12, 0, 3, 3, 0, 3, 3, 6, 9, 9, 0, 3, 6, 0, 9, 3, 12, 3, 3, 0, 6, 9, 0, 3, 6, 12, 3, 3, 0, 3

Offset: 1

Bernard Schott, Nov 27 2019

nonn

Some results coming from the Alarcon and Duval reference.
For n = 0, there are infinitely many solutions because every triple (k,k,k) with k >= 0 satisfies the equation.
a(n) = 0 iff 3 divides n and 9 doesn't divide n (equivalent to n is in A016051).
When n belongs to A074232 (complement of A016051), a(n) is always a multiple of 3 because
1) if (a,a,b) [resp. (a,b,b)] with a < b is a primitive solution, then these triples generate 3 solutions with the permutations (a,a,b), (a,b,a), (b,a,a), [resp. (a,b,b), (b,b,a), (b,a,b)] and,
2) if (a,b,c) with a < b < c is a primitive solution, then this triple generates 6 solutions with the permutations (a,b,c), (b,c,a), (c,a,b), (a,c,b), (c,b,a), (b,a,c).
For prime p <> 3, a(p) = a(2*p) = 3.
An inequality: (n/4)^(1/3) <= max(x, y, z) <= (n+2)/3.
This sequence is unbounded.
A261029 gives the number of triples without counting the permutations and, in link, a list of primitive triples up to n = 2000.

			3^3+2^3+2^3-3*2*2*3 = 7 so (3,2,2), (2,2,3) and (2,3,2) are solutions and a(7) = 3.
When n=35, (0,1,3) is a primitive solution that generates 6 solutions and (9,9,10) is another primitive solution that generates 3 solutions, so a(35)=6+3=9 (see comments).

Guy Alarcon and Yves Duval, TS: Préparation au Concours Général, RMS, Collection Excellence, Paris, 2010, chapitre 9, Problème: étude d'une équation diophantienne cubique, pages 137-138 and 147-152.

Vladimir Shevelev, Representation of positive integers by the form x^3+y^3+z^3-3xyz, arXiv:1508.05748 [math.NT], 2015.

Cf. A016051, A074232.
Cf. A261029 (primitive triples without the permutations).
Cf. A050787, A050791, A212420 (other cubic Diophantine equations).

a[n_] := Length@ Solve[x^3 + y^3 + z^3 - 3 x y z == n && x >= 0 && y >= 0 && z >= 0, {x, y, z}, Integers]; Array[a, 85] (* Giovanni Resta, Nov 28 2019 *)

If n = 3*k + 1, then (k, k, k+1) is a solution for k >= 0.
If n = 3*k - 1, then (k, k, k-1) is a solution for k >= 1.
If n = 9*k, then (k-1, k, k+1) is a solution for k >= 1.
If n = k^3, then (k, 0, 0) is a solution for k >= 0.
If n = 2*k^3, then (k, k, 0) is a solution for k >= 0.

More terms from Giovanni Resta, Nov 28 2019

cp's OEIS Frontend

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

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A330013 a(n) is the number of solutions with nonnegative (x,y,z) to the cubic Diophantine equation x^3+y^3+z^3 - 3xy*z = n.

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cp's OEIS Frontend

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A330013 a(n) is the number of solutions with nonnegative (x,y,z) to the cubic Diophantine equation x^3+y^3+z^3 - 3*x*y*z = n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A330013 a(n) is the number of solutions with nonnegative (x,y,z) to the cubic Diophantine equation x^3+y^3+z^3 - 3xy*z = n.