A234319 - OEIS

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.

%I #58 Sep 04 2021 14:52:41

%S 0,3,25,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14,-15,-16,-17,-18,-19,

%T -20,-21,-22,-23,-24,-25,-26,-27,-28,-29,-30,-31,-32,-33,-34,-35,-36,

%U -37,-38,-39,-40,-41,-42,-43,-44,-45,-46,-47,-48,-49,-50,-51,-52,-53,-54

%N 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.

%C a(n) is the 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.

%C In 1879 Dostor gave all solutions for n = 2. In particular, a(2) = 25.

%C In 1906 Collignon proved that no solution exists for n = 3 and 4, so a(3) = -3 and a(4) = -4.

%C 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.

%D 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.

%H L. E. Dickson, <a href="http://books.google.com/books?id=9LQqAwAAQBAJ&pg=PA564&lpg=PA564&dq=dickson+%22noted+that+there+is+no+positive+integral+solution+of%22%22&source=bl&ots=rZeMQFzLta&sig=FOsVXdiPjunRzxcDFP819lUoYEY&hl=en&sa=X&ei=2qGlU5HWA6ic8gGYvYCABQ&ved=0CB8Q6AEwAA#v=onepage&q=dickson%20%22noted%20that%20there%20is%20no%20positive%20integral%20solution%20of%22%22&f=false">History of the Theory of Numbers, II</a>, p. 564.

%H Georges Dostor, <a href="https://www.digitale-sammlungen.de/en/view/bsb11390652?page=372">Question sur les nombres</a>, Archiv der Mathematik und Physik, 64 (1879), 350-352.

%H Simon Felten and Stefan Müller-Stach, <a href="http://arxiv.org/abs/1312.5943">A diophantine equation for sums of consecutive like powers</a>, arXiv:1312.5943 [math.NT], 2013-2015; Elem. Math., 70 (2015), 117-124. doi: 10.4171/EM/284

%H Greg Frederickson, <a href="http://www.jstor.org/stable/27765930">Casting Light on Cube Dissections</a>, Math. Mag., 82 (2009), 323-331.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(0) = A059270(0) = A059255(0).

%F a(1) = A059270(1) = A230718(1).

%F a(2) = A059255(2) = A230718(2).

%F a(n) = -n for n > 2.

%F G.f.: x*(27*x^3-50*x^2+19*x+3) / (x-1)^2. - _Colin Barker_, Apr 23 2014

%e 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.

%e 1^1 + 2^1 = 3 = 3^1 is minimal for n = 1, so a(1) = 3.

%e 3^2 + 4^2 = 25 = 5^2 is minimal for n = 2, so a(2) = 25.

%t 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 *)

%o (PARI) Vec(x*(27*x^3-50*x^2+19*x+3)/(x-1)^2 + O(x^100)) \\ _Colin Barker_, Apr 23 2014

%Y Cf. A059255, A059270, A222716, A230718.

%K sign,easy

%O 0,2

%A _Jonathan Sondow_, Dec 23 2013