The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square. 10
 1, 3, 7, 9, 11, 13, 15, 17, 18, 20, 21, 22, 25, 27, 28, 30, 32, 38, 44, 50, 52, 55, 58, 60, 64, 65, 67, 73, 74, 76, 83, 87, 91, 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200, 204 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS That is, numbers n such that Sum_{i=n..k} i^2 is a square for some k > n. The paper by Bremner, Stroeker, and Tzanakis describes how they found all n <= 100 by solving elliptic curves. Their solutions are the same as the terms in this sequence. They also show that there are only a finite number of sums of squares beginning with n^2 that sum to a square. For example, starting with 3^2, there are only 3 ways to sum consecutive squares to produce a square: 3^2 + 4^2, 3^2 + ... + 580^2, and 3^2 + ... + 963^2. See A184762, A184763, A184885, and A184886 for more results from their paper. This sequence is more difficult than A001032, which has the possible lengths of the sequences of consecutive squares that sum to a square. Be careful adding terms to this sequence; a simple search may miss some terms. An elliptic curve needs to be solved for each number. It is conjectured that the sequence continues 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200. - Jean-François Alcover, Sep 17 2013. Conjecture confirmed (see the Schoenfield link below). - Jon E. Schoenfield, Nov 22 2013 LINKS A. Bremner, R. J. Stroeker, and N. Tzanakis, On Sums of Consecutive Squares, J. Number Theory 62 (1997), 39-70. K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power Masoto Kuwata, Jaap Top, An elliptic surface related to sums of consecutive squares, Exposition. Math. 12 (1994) 181-192 FORMULA Numbers n such that A075404(n) > 0. EXAMPLE 30 is in the sequence because 30^2 + 31^2 + 32^2 + ... + 197^2 + 198^2 = 1612^2. MATHEMATICA jmax = jmax = 10^5; jmax[n_ /; n > 91] = 10^6; jmax[_] = 10^4; Reap[For[n = 1, n <= 200, n++, s = n^2; For[j = n+1, j <= jmax[n], j++, s += j^2; If[IntegerQ[Sqrt[s]], Sow[n]; Print[n, "(", j, ", ", Sqrt[s], ")"]; Break[]]]]][[2, 1]] (* Jean-François Alcover, Sep 17 2013, translated and adapted from Pari *) PROG (PARI)for(n=1, 100, s=n^2; for(j=n+1, 999999, s+=j^2; if(issquare(s), print1(n, "(", j, ", ", sqrtint(s), ")"); break()))) CROSSREFS Cf. A180259, A180273, A180274, A180465. Cf. A075404, A075405, A075406. Sequence in context: A179197 A097270 A109802 * A165631 A245586 A005818 Adjacent sequences:  A180439 A180440 A180441 * A180443 A180444 A180445 KEYWORD nonn,nice AUTHOR Zhining Yang, Jan 19 2011 EXTENSIONS Example simplified by Jon E. Schoenfield, Sep 18 2013 More terms from Jon E. Schoenfield, Nov 22 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 16 05:56 EDT 2022. Contains 353693 sequences. (Running on oeis4.)