The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A186699 Numbers n such that there are n numbers in arithmetic progression whose squares sum to a perfect square. 3
 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: A101255 A356181 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 6 22:19 EDT 2023. Contains 363151 sequences. (Running on oeis4.)