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Search: id:A001032
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| A001032 |
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Numbers n such that sum of squares of n consecutive integers >= 1 is a square.
(Formerly M1996 N0787)
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+0
7
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| 1, 2, 11, 23, 24, 26, 33, 47, 49, 50, 59, 73, 74, 88, 96, 97, 107, 121, 122, 146, 169, 177, 184, 191, 193, 194, 218, 239, 241, 242, 249, 289, 297, 299, 311, 312, 313, 337, 338, 347, 352, 361, 362, 376, 383, 393, 407, 409, 431, 443, 457, 458, 479, 481, 491, 506
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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It was shown by Watson (and again by Ljunggren) that if 0^2 + 1^2 + ... + r^2 is a square then r = 0, 1 or 24.
The terms up to 1391 are == 0, 1, 2, 9, 11, 16, 23 mod 24. Start number is in A007475(n). Square root of sum is in A076215(n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Nov 04 2002
The solutions in the case n=2 are in A001652 or A082291.
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REFERENCES
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U. Alfred, Consecutive integers whose sum of squares is a perfect square, Math. Mag., 37 (1964), 19-32.
L. Beeckmans, Squares expressible as sum of consecutive squares, Amer. Math. Monthly, 101 (1994), 437-442.
W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tid. 34 (1952), 65-72.
S. Philipp, Note on consecutive integers whose sum of squares is a perfect square, Math. Mag., 37 (1964), 218-220.
G. N. Watson, The problem of the square pyramid, Messenger Math. 48, 1-22, 1918.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..128
Kevin Brown, Sum of Consecutive Nth Powers Equals an Nth Power
Eric Weisstein's World of Mathematics, Cannonball Problem
Index entries for sequences related to sums of squares
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EXAMPLE
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3^2 + 4^2 = 5^2, with two consecutive terms, so 2 is in the sequence.
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CROSSREFS
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Cf. A007475, A076215.
Cf. A097812 (n^2 is the sum of two or more consecutive squares).
Sequence in context: A118594 A018351 A004642 this_sequence A066079 A045386 A084354
Adjacent sequences: A001029 A001030 A001031 this_sequence A001033 A001034 A001035
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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EXTENSIONS
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Corrected by T. D. Noe (noe(AT)sspectra.com), Aug 25 2004
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