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 A234319 Smallest sum of n-th powers of k+1 consecutive positive integers that equals the sum of n-th powers of the next k consecutive integers, or -n if none. 2
 0, 3, 25, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30, -31, -32, -33, -34, -35, -36, -37, -38, -39, -40, -41, -42, -43, -44, -45, -46, -47, -48, -49, -50, -51, -52, -53, -54 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = smallest solution to m^n + (m+1)^n + … + (m+k)^n = (m+k+1)^n + (m+k+2)^n + ... + (m+2*k)^n, or -n if no solution. In 1879 Dostor gave all solutions for n = 2. In particular, a(2) = 25. In 1906 Collignon proved that no solution exists for n = 3 and 4, so a(3) = -3 and a(4) = -4. In 2013 Felten and Müller-Stach claimed to prove that no solution exists when n > 2, so if their proof is correct, a(n) = -n for n >= 3. REFERENCES Edouard Collignon, Note sur la résolution en entiers de m^2 + (m-r)^2 + … + (m-kr)^2 = (m+r)^2 + ... + (m+kr)^2, Sphinx-Oedipe, 1 (1906-1907), 129-133. Georges Dostor, Question sur les nombres, Archiv der Mathematik und Physik, 64 (1879), 350-352. LINKS L. E. Dickson, History of the Theory of Numbers, II, p. 564. Simon Felten and Stefan Müller-Stach, A diophantine equation for sums of consecutive like powers, arXiv:1312.5943 [math.NT], Elem. Math., 70 (2015), 117-124. doi: 10.4171/EM/284 Greg Frederickson, Casting Light on Cube Dissections, Math. Mag., 82 (2009), 323-331. Index entries for linear recurrences with constant coefficients, signature (2,-1). FORMULA a(0) = A059270(0) = A059255(0). a(1) = A059270(1) = A230718(1). a(2) = A059255(2) = A230718(2). a(n) = -n for n > 2. G.f.: x*(27*x^3-50*x^2+19*x+3) / (x-1)^2. - Colin Barker, Apr 23 2014 EXAMPLE m^0 + (m+1)^0 + ... + (m+k)^0 = k+1 > k = (m+k+1)^0 + (m+k+2)^0 + ... + (m+2*k)^0 for m > 0, so a(0) = -0 = 0. 1^1 + 2^1 = 3 = 3^1 is minimal for n = 1, so a(1) = 3. 3^2 + 4^2 = 25 = 5^2 is minimal for n = 2, so a(2) = 25. MATHEMATICA CoefficientList[Series[x*(27*x^3 - 50*x^2 + 19*x + 3)/(x - 1)^2, {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 21 2014 *) PROG (PARI) Vec(x*(27*x^3-50*x^2+19*x+3)/(x-1)^2 + O(x^100)) \\ Colin Barker, Apr 23 2014 CROSSREFS Cf. A059255, A059270, A222716, A230718. Sequence in context: A297533 A080202 A300945 * A224873 A085836 A073916 Adjacent sequences:  A234316 A234317 A234318 * A234320 A234321 A234322 KEYWORD sign,easy AUTHOR Jonathan Sondow, Dec 23 2013 STATUS approved

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Last modified February 19 01:03 EST 2020. Contains 332028 sequences. (Running on oeis4.)