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A097812
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Numbers n such that n^2 is the sum of two or more consecutive squares.
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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.
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LINKS
| K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power
Index entries for sequences related to sums of squares
Donovan Johnson, Table of n, a(n) for n = 1..5077
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EXAMPLE
| 29 is in this sequence because 20^2 + 21^2 = 29^2.
Donovan Johnson, Feb 19 2011: 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.
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MATHEMATICA
| g[m0_, m1_] := (1-m0+m1)(-m0+2m0^2+m1+2m0 m1+2m1^2)/6; lst={}; Do[n=g[m0, m1]^(1/2); If[IntegerQ[n], Print[{n, m0, m1, m1-m0+1}]; AppendTo[lst, n]], {m1, 2, 2000}, {m0, m1-1, 1, -1}]; Union[lst]
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CROSSREFS
| Cf. A097811 (n^3 is the sum of consecutive cubes).
Cf. A001032, A151557.
Sequence in context: A031394 A103094 A108928 * A176333 A100559 A087348
Adjacent sequences: A097809 A097810 A097811 * A097813 A097814 A097815
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Aug 25 2004
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