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A097812
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Numbers n such that n^2 is the sum of two or more consecutive positive squares.
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8
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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29 is in this sequence because 20^2 + 21^2 = 29^2.
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)
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A097811 (n^3 is the sum of consecutive cubes).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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