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 A259060 Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 27 17:53 EDT 2023. Contains 365714 sequences. (Running on oeis4.)