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A001032 Numbers n such that sum of squares of n consecutive integers >= 1 is a square.
(Formerly M1996 N0787)
17
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; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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, Nov 04 2002

The solutions in the case n=2 are in A001652 or A082291.

For k>5 and k = 1 or 5 (mod 6), it appears that all k^2 are here. When n is not a square, the solution to problem 6552 shows that there are an infinite number of sums of n consecutive squares that equal a square. There are only a finite number when n is a square. For example, the only sum having 49 terms is 25^2+...+73^2 = 357^2. - T. D. Noe, Jan 20, 2011

In the previous comment, "it appears" can be removed because the k^2 squares beginning at (k^2+1)(k^2-25)/48 sum to a square. - Thomas Andrews, Feb 14 2011

See A180442 for the complementary problem of finding numbers n such that there are consecutive squares beginning with n^2 that sum to a square.

Contribution from Thomas Andrews, Feb 22 2011: (Start)

Elementary necessary conditions for n to be in this sequence:

  1. If n=s^2b where b is square-free, then:

     a. If s is divisible by 3 then b is divisible by 3.

     b. If s is divisible by 2, then b is divisible by 2.

     c. If b is divisible by 3, then b = 6 (mod 9)

     d. b only has prime factors p where 3 is a square, modulo p.  (So, p=2, p=3, or p=12k+/-1)

  2.

     a. If n+1 is divisible by 3, then (n+1)/3 is the sum of two perfect squares.

     b. If n+1 is not divisible by 3, then n+1 is the sum of two perfect squares

The smallest number which satisfies these conditions which is not in this sequence is 842.

These conditions can be used to show the conjecture by Ralf Stephan, above, that all the numbers are == 0, 1, 2, 9, 11, 16, or 23 (mod 24.) (End)

REFERENCES

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.

Squares Expressible as a Sum of n Consecutive Squares, Advanced Problem 6552, proposed by M. Laub, Amer. Math. Monthly 97 (1990), 622-625.

LINKS

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

EXAMPLE

3^2 + 4^2 = 5^2, with two consecutive terms, so 2 is in the sequence.

MATHEMATICA

(* An empirical recomputation, assuming Ralf Stephan's conjecture *) nmax = 600; min[_](* minimum start number *) = 1; max[_](* maximum start number *) = 10^5; min[457(* the first not-so-easy term *)] = 10^7; min[577] = 10^5; min[587] = 10^7; max[457] = max[577] = max[587] = Infinity; okQ[n_ /; ! MemberQ[{0, 1, 2, 9, 11, 16, 23}, Mod[n, 24]]] = False; okQ[n_] := For[m = min[n], m < max[n], m++, If[IntegerQ[ r = Sqrt[1/6*n*(1 + 6*m^2 + 6*m*(n - 1) - 3*n + 2*n^2)]], Return[True]]]; nmr = Reap[k = 1; Do[If[okQ[n] === True, Print["a(", k, ") = ", n, ", start nb = A007475(", k, ") = ", m, ", sqrt(sum) = A076215(", k, ") = ", r]; k++; Sow[{n, m, r}]], {n, 1, nmax}]][[2, 1]]; A001032 = nmr[[All, 1]]; A007475 = nmr[[All, 2]]; A076215 = nmr[[All, 3]] (* Jean-Fran├žois Alcover, Sep 09 2013 *)

CROSSREFS

Cf. A007475, A076215, A151577.

Cf. A097812 (n^2 is the sum of two or more consecutive squares).

Sequence in context: A018351 A004642 A185545 * A045386 A084354 A066725

Adjacent sequences:  A001029 A001030 A001031 * A001033 A001034 A001035

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected by T. D. Noe, Aug 25 2004

STATUS

approved

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Last modified July 31 17:29 EDT 2014. Contains 245085 sequences.