login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

A097812
Numbers n such that n^2 is the sum of two or more consecutive positive squares.
8
5, 29, 70, 77, 92, 106, 138, 143, 158, 169, 182, 195, 245, 253, 274, 357, 385, 413, 430, 440, 495, 531, 650, 652, 655, 679, 724, 788, 795, 985, 1012, 1022, 1055, 1133, 1281, 1365, 1397, 1518, 1525, 1529, 1546, 1599, 1612, 1786, 1828, 2205, 2222, 2257, 2372
OFFSET
1,1
COMMENTS
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.
EXAMPLE
29 is in this sequence because 20^2 + 21^2 = 29^2.
Contribution from Donovan Johnson, Feb 19 2011: (Start)
For seven terms < (10^15)^(1/2), the square is a sum in two different ways:
143^2 = 7^2 + ... + 39^2 = 38^2 + ... + 48^2.
2849^2 = 294^2 + ... + 367^2 = 854^2 + ... + 864^2.
208395^2 = 2175^2 + ... + 5199^2 = 29447^2 + ... + 29496^2.
2259257^2 = 9401^2 + ... + 25273^2 = 26181^2 + ... + 32158^2.
6555549^2 = 41794^2 + ... + 58667^2 = 87466^2 + ... + 92756^2.
11818136^2 = 10898^2 + ... + 74906^2 = 29929^2 + ... + 76392^2.
19751043^2 = 39301^2 + ... + 107173^2 = 249217^2 + ... + 255345^2. (End)
MATHEMATICA
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]
CROSSREFS
Cf. A097811 (n^3 is the sum of consecutive cubes).
Sequence in context: A301858 A293174 A108928 * A176333 A100559 A224498
KEYWORD
nonn
AUTHOR
T. D. Noe, Aug 25 2004
EXTENSIONS
Name edited by Altug Alkan, Dec 07 2015
STATUS
approved