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 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). 4
 504, 8721, 65472, 312375, 1119528, 3293829, 8388096, 19131147, 39999000, 77947353, 143325504, 250991871, 421651272, 683434125, 1073737728, 1641349779, 2448874296, 3575480097, 5119992000, 7204344903, 9977420904, 13619289621, 18345871872, 24414046875 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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). 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). The trivial solutions with M < 1 and b < 2 are not considered here. LINKS Vladimir Pletser, Table of n, a(n) for n = 2..50000 Vladimir Pletser, File Triplets (M,b,c) for M=2n^2 Vladimir Pletser, Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares, Research Gate, 2015. Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015. R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307. Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1). FORMULA a(n) = n^3(4n^4 - 1). 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 EXAMPLE For n=2, M(n)=8, b(n)=28, c(n)=504. See "File Triplets (M,b,c) for M=2n^2" link. MAPLE restart: for n from 2 to 50000 do a:= n^3*(4*n^4 - 1): print (a); end do: MATHEMATICA f[n_] := n^3 (4 n^4 - 1); Rest@Array[f, 32] (* Michael De Vlieger, Jan 28 2015 *) PROG (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 (Magma) [n^3*(4*n^4 - 1): n in [2..30]]; // Vincenzo Librandi, Feb 19 2015 CROSSREFS Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253681, A253707, A253709, A002593, A253725. Sequence in context: A263286 A061124 A141145 * A166763 A012829 A013973 Adjacent sequences: A253721 A253722 A253723 * A253725 A253726 A253727 KEYWORD nonn,easy AUTHOR Vladimir Pletser, Jan 10 2015 STATUS approved

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Last modified April 17 15:33 EDT 2024. Contains 371764 sequences. (Running on oeis4.)