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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
9
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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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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
K. S. 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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N. J. A. Sloane (njas(AT)research.att.com).
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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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