A097812 - OEIS

%I #23 Dec 18 2015 11:57:28

%S 5,29,70,77,92,106,138,143,158,169,182,195,245,253,274,357,385,413,

%T 430,440,495,531,650,652,655,679,724,788,795,985,1012,1022,1055,1133,

%U 1281,1365,1397,1518,1525,1529,1546,1599,1612,1786,1828,2205,2222,2257,2372

%N Numbers n such that n^2 is the sum of two or more consecutive positive squares.

%C These numbers were found by exhaustive search. The sums are not unique; for n = 143, there are two representations. The Mathematica code prints n, the range of squares in the sum and the number of squares in the sum. Because the search included sums of all squares up to 2000, this sequence is complete up to 2828.

%H Donovan Johnson, <a href="/A097812/b097812.txt">Table of n, a(n) for n = 1..5077</a>

%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath147.htm">Sum of Consecutive Nth Powers Equals an Nth Power</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%e 29 is in this sequence because 20^2 + 21^2 = 29^2.

%e Contribution from _Donovan Johnson_, Feb 19 2011: (Start)

%e For seven terms < (10^15)^(1/2), the square is a sum in two different ways:

%e 143^2 = 7^2 + ... + 39^2 = 38^2 + ... + 48^2.

%e 2849^2 = 294^2 + ... + 367^2 = 854^2 + ... + 864^2.

%e 208395^2 = 2175^2 + ... + 5199^2 = 29447^2 + ... + 29496^2.

%e 2259257^2 = 9401^2 + ... + 25273^2 = 26181^2 + ... + 32158^2.

%e 6555549^2 = 41794^2 + ... + 58667^2 = 87466^2 + ... + 92756^2.

%e 11818136^2 = 10898^2 + ... + 74906^2 = 29929^2 + ... + 76392^2.

%e 19751043^2 = 39301^2 + ... + 107173^2 = 249217^2 + ... + 255345^2. (End)

%t g[m0_, m1_] := (1 - m0 + m1)(-m0 + 2m0^2 + m1 + 2m0 m1 + 2m1^2)/6; A097812 = {}; Do[n = g[m0, m1]^(1/2); If[IntegerQ[n], Print[{n, m0, m1, m1 - m0 + 1}]; AppendTo[A097812, n]], {m1, 2, 2000}, {m0, m1 - 1, 1, -1}]; Union[A097812]

%Y Cf. A097811 (n^3 is the sum of consecutive cubes).

%Y Cf. A001032, A151557.

%K nonn

%O 1,1

%A _T. D. Noe_, Aug 25 2004

%E Name edited by _Altug Alkan_, Dec 07 2015