

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
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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.


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..5077
K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power
Index entries for sequences related to sums of squares


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).
Cf. A001032, A151557.
Sequence in context: A301858 A293174 A108928 * A176333 A100559 A224498
Adjacent sequences: A097809 A097810 A097811 * A097813 A097814 A097815


KEYWORD

nonn


AUTHOR

T. D. Noe, Aug 25 2004


EXTENSIONS

Name edited by Altug Alkan, Dec 07 2015


STATUS

approved



