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A186699 Numbers n such that there are n numbers in arithmetic progression whose squares sum to a perfect square. 1
1, 2, 4, 9, 11, 16, 23, 24, 25, 26, 33, 36, 47, 49, 50, 52, 59, 64, 73, 74, 81, 88, 96, 97, 100, 107, 121, 122, 144, 146, 148, 169, 177, 184, 191, 193, 194, 196, 218, 225, 239, 241, 242, 244, 249, 256, 276, 289, 292, 297, 299, 311, 312, 313, 324, 337, 338 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A positive integer n is in this sequence if and only if there is a solution to the Pell-like equation x^2-ny^2=d^2(n-1)n(n+1)/3 for some x,y,d integers.

A positive integer n is in this sequence if and only if it can be written in the form: (u^2-3w^2)/(v^2+3w^2), with u,v,w integers and gcd(v,w)=1. This can also be written as a  n(v^2) + 3(n+1)(w^2) = z^2.

If n is in this sequence, then we can find an arithmetic progression of *positive* integers which satisfy this equation. (The description above does not require the sequence to be positive.)

By using the method of Legendre to find whether there exists rational numbers r,s on the curve nr^2 + 3(n+1)s^2 = 1, we get the following necessary and sufficient conditions on n:

  A. Factor n=a^2b, with b squarefree, then

    1. If 3 does not divide b(n+1), then b ≅ 1 (mod 3)

    2. If b is divisible by 3, then b ≅ 6 (mod 9)

    3. 3 is a square (mod b.)

  B.

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

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

When n is a perfect square, we can use the arithmetic sequence starting at m=(3n+2)(sqrt(n)-1)/2 + 6 and common difference 6.

LINKS

Table of n, a(n) for n=1..57.

Thomas Andrews, Article about this and related problems

Thomas Andrews, Initial Terms

Index entries for sequences related to sums of squares

EXAMPLE

For n=4, (13,19,25,31) is an arithmetic progression of length 4, and 13^2+19^2+25^2+31^2 = 46^2, so 4 is in the sequence.

CROSSREFS

Cf. A134419 is a subsequence.

Sequence in context: A190494 A101255 A047348 * A093859 A266257 A115905

Adjacent sequences:  A186696 A186697 A186698 * A186700 A186701 A186702

KEYWORD

nonn

AUTHOR

Thomas Andrews, Feb 25 2011, Mar 12 2011

STATUS

approved

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Last modified December 16 09:06 EST 2019. Contains 330020 sequences. (Running on oeis4.)