cp's OEIS Frontend

A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square.

%I A180442 #49 Jan 14 2024 14:28:50
%S A180442 1,3,7,9,11,13,15,17,18,20,21,22,25,27,28,30,32,38,44,50,52,55,58,60,
%T A180442 64,65,67,73,74,76,83,87,91,103,104,106,112,115,117,119,121,124,128,
%U A180442 129,131,132,137,140,142,146,158,168,170,172,175,178,181,183,192,193,197,199,200,204
%N A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square.
%C A180442 That is, numbers n such that Sum_{i=n..k} i^2 is a square for some k > n.
%C A180442 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.
%C A180442 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.
%C A180442 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
%H A180442 Jon E. Schoenfield, <a href="/A180442/b180442.txt">Table of n, a(n) for n = 1..123 (includes a derivation of the elliptic curves and Magma code used to find the terms)</a>
%H A180442 A. Bremner, R. J. Stroeker, N. Tzanakis, <a href="https://doi.org/10.1006/jnth.1997.2040">On Sums of Consecutive Squares</a>, J. Number Theory 62 (1997), 39-70.
%H A180442 K. S. Brown, <a href="http://www.mathpages.com/home/kmath147.htm">Sum of Consecutive Nth Powers Equals an Nth Power</a>
%H A180442 Masoto Kuwata, Jaap Top, <a href="http://www.math.rug.nl/~top/toku2.pdf">An elliptic surface related to sums of consecutive squares</a>, Exposition. Math. 12 (1994) 181-192
%F A180442 Numbers n such that A075404(n) > 0.
%e A180442 30 is in the sequence because 30^2 + 31^2 + 32^2 + ... + 197^2 + 198^2 = 1612^2.
%t A180442 jmax[11] = jmax[74] = 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 *)
%o A180442 (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())))
%Y A180442 Cf. A180259, A180273, A180274, A180465.
%Y A180442 Cf. A075404, A075405, A075406.
%K A180442 nonn,nice
%O A180442 1,2
%A A180442 _Zhining Yang_, Jan 19 2011
%E A180442 Example simplified by _Jon E. Schoenfield_, Sep 18 2013
%E A180442 More terms from _Jon E. Schoenfield_, Nov 22 2013