A259060 Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes.

6426, 7900, 9614, 11592, 13858, 16436, 19350, 22624, 26282, 30348, 34846, 39800, 45234, 51172, 57638, 64656, 72250, 80444, 89262, 98728, 108866, 119700, 131254, 143552, 156618, 170476, 185150, 200664, 217042, 234308, 252486, 271600, 291674

Offset

0,1

Comments

This is the second part of Exercise 229 in Sierpiński's problem book. See p. 20, and p. 110 for the solution. He uses the identity (n-8)^3 + (n-1)^3 + (n+1)^3 + (n+8)^3 = 4*n^3 + 390 = (n-7)^3 + (n-4)^3 + (n+4)^3 + (n+7)^3, for n >= 9.

Here n is replaced by n + 9: (n+1)^3 + (n+8)^3 + (n+10)^3 + (n+17)^3 = 4*n^3 + 108*n^2 + 1362*n + 6426 = (n+2)^3 + (n+5)^3 + (n+13)^3 + (n+16)^3, for n >= 0.

There may be other numbers with this properties.

Because the summands have no common factor > 1 each of these two representations is called primitive. - Wolfdieter Lang, Aug 20 2015

References

W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970.

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = (2*(n+9))*(2*n^2+36*n+357) = 2*A261241(n), n >= 0. See the comment for the sum of four distinct cubes in two different ways.

O.g.f.: 2*(3213 - 8902*x + 8285*x^2 - 2584*x^3) / (1-x)^4.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Aug 13 2015

Example

a(0) = 6426 = 1^3 + 8^3 + 10^3 + 17^3 = 2^3 + 5^3 + 13^3 + 16^3.
a(1) = 7900 = 2^3 + 9^3 + 11^3 + 18^3 = 3^3 + 6^3 + 14^3 + 17^3.

Mathematica

CoefficientList[Series[2 (3213 - 8902 x + 8285 x^2 - 2584 x^3)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 13 2015 *)
LinearRecurrence[{4, -6, 4, -1}, {6426, 7900, 9614, 11592}, 40] (* Harvey P. Dale, Sep 30 2016 *)

Prog

(Magma) [(2*(n+9))*(2*n^2+36*n+357): n in [0..50]] /* or */ I:=[6426, 7900, 9614, 11592]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 13 2015

Crossrefs

Cf. A261241, A259058 (squares).

Sequence in context: A270056 A031578 A252612 * A259078 A104374 A222554

Adjacent sequences: A259057 A259058 A259059 * A259061 A259062 A259063

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 12 2015

Status

approved