cp's OEIS Frontend

A253724 Numbers c(n) whose squares are equal to the sums of a number M(n) of consecutive cubed integers b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2, starting at b(n) (A002593) for M(n) being twice a squared integer (A001105).

%I A253724 #41 Sep 08 2022 08:46:10
%S A253724 504,8721,65472,312375,1119528,3293829,8388096,19131147,39999000,
%T A253724 77947353,143325504,250991871,421651272,683434125,1073737728,
%U A253724 1641349779,2448874296,3575480097,5119992000,7204344903,9977420904,13619289621,18345871872,24414046875
%N A253724 Numbers c(n) whose squares are equal to the sums of a number M(n) of consecutive cubed integers b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2, starting at b(n) (A002593) for M(n) being twice a squared integer (A001105).
%C A253724 Numbers c(n) such that b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2 has nontrivial solutions over the integers for M(n) being twice a squared integer (A001105) and b(n)=(A002593).
%C A253724 If M is twice a squared integer, there always exists at least one nontrivial solution for the sum of M consecutive cubed integers starting at b^3 and equaling to a squared integer c^2. For n>=1, M(n)= 2n^2 (A001105), b(n) = M(M-1)/2 = n^2(2n^2 - 1) (A002593), and c(n)= sqrt(M/2) (M(M^2-1)/2)= n^3(4n^4 - 1) (this sequence).
%C A253724 The trivial solutions with M < 1 and b < 2 are not considered here.
%H A253724 Vladimir Pletser, <a href="/A253724/b253724.txt">Table of n, a(n) for n = 2..50000</a>
%H A253724 Vladimir Pletser, <a href="/A253724/a253724_1.txt">File Triplets (M,b,c) for M=2n^2</a>
%H A253724 Vladimir Pletser, <a href="http://www.researchgate.net/profile/Vladimir_Pletser/publication/271272786">Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares</a>, Research Gate, 2015.
%H A253724 Vladimir Pletser, <a href="http://arxiv.org/abs/1501.06098">General solutions of sums of consecutive cubed integers equal to squared integers</a>, arXiv:1501.06098 [math.NT], 2015.
%H A253724 R. J. Stroeker, <a href="http://www.numdam.org/item?id=CM_1995__97_1-2_295_0">On the sum of consecutive cubes being a perfect square</a>, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.
%H A253724 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A253724 a(n) = n^3(4n^4 - 1).
%F A253724 G.f.: -3*x^2*(x^7-8*x^6+27*x^5-216*x^4-1521*x^3-3272*x^2-1563*x-168) / (x-1)^8. - _Colin Barker_, Jan 14 2015
%e A253724 For n=2, M(n)=8, b(n)=28, c(n)=504.
%e A253724 See "File Triplets (M,b,c) for M=2n^2" link.
%p A253724 restart: for n from 2 to 50000 do a:= n^3*(4*n^4 - 1): print (a); end do:
%t A253724 f[n_] := n^3 (4 n^4 - 1); Rest@Array[f, 32] (* _Michael De Vlieger_, Jan 28 2015 *)
%o A253724 (PARI) Vec(-3*x^2*(x^7-8*x^6+27*x^5-216*x^4-1521*x^3-3272*x^2-1563*x-168)/(x-1)^8 + O(x^100)) \\ _Colin Barker_, Jan 14 2015
%o A253724 (Magma) [n^3*(4*n^4 - 1): n in [2..30]]; // _Vincenzo Librandi_, Feb 19 2015
%Y A253724 Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253681, A253707, A253709, A002593, A253725.
%K A253724 nonn,easy
%O A253724 2,1
%A A253724 _Vladimir Pletser_, Jan 10 2015