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A123204 Numbers that can be written as the sum of two squares in three ways, using three consecutive squares. 0
1, 25, 1105, 12025, 21025, 66625, 252601, 292825, 751825, 1026745, 1671865, 1892185, 4210945, 4534945, 8529625, 8958625, 10251025, 16040401, 24019801, 28404025, 29138425, 29604625, 47859265, 51396865, 53438905, 62747425 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Numbers can be parameterized as follows: choose an odd number a and k a divisor of (a^2+1)/2, k <= sqrt((a^2+1)/2) (otherwise you will get a duplicate value with a negative number being squared) and let d = a/k * (a^2+3ak+2k^2+1) / k. Then the three consecutive squares are (d-1)/2, (d+1)/2 and (d+3)/2. The squares added to them are (d/a + a) / 2, (d/a - a) / 2 and ((d+2)/(a+2k) - (a+2k))/2. All members of this sequence are congruent to 1 or 25 (mod 120); all are the product exclusively of primes congruent to 1 (mod 4). No number can written as the sum of two squares using four consecutive squares.

LINKS

Table of n, a(n) for n=1..26.

EXAMPLE

1105 = 31^2 + 12^2 = 32^2 + 9^2 = 33^2 + 4^2, using the consecutive values 31,32,33. 1 is included for the triple -1,0,1.

CROSSREFS

Cf. A002144, A001481.

Sequence in context: A193121 A246923 A066852 * A012508 A112102 A012799

Adjacent sequences:  A123201 A123202 A123203 * A123205 A123206 A123207

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Oct 04 2006

STATUS

approved

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Last modified May 15 12:37 EDT 2021. Contains 343920 sequences. (Running on oeis4.)