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A273293 Numbers n such that n and n^2 are the sums of two nonzero squares in exactly two ways. 0
50, 200, 338, 450, 578, 800, 1352, 1682, 1800, 2312, 2450, 2738, 3042, 3200, 3362, 4050, 5202, 5408, 5618, 6050, 6728, 7200, 7442, 9248, 9800, 10658, 10952, 12168, 12800, 13448, 15138, 15842, 16200, 16562, 18050, 18818, 20402, 20808, 21632, 22050, 22472, 23762, 24200, 24642, 25538 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If n is the sum of 2 nonzero squares in exactly 2 ways, then n = a^2 + b^2 = c^2 + d^2 where (a, b), (c, d) are distinct and a, b, c, d are nonzero. For n^2;

n^2 = (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2,

n^2 = (a^2 + b^2)*(c^2 + d^2) = (a*d + b*c)^2 + (a*c - b*d)^2,

n^2 = (a^2 + b^2)*(a^2 + b^2) = (a^2 - b^2)^2 + (2*a*b)^2,

n^2 = (c^2 + d^2)*(c^2 + d^2) = (c^2 - d^2)^2 + (2*c*d)^2.

Note that if n is of the form 2*m^2 where m is nonzero integer, then the first two representations will be the same and one of the last two identities will not be the sum of two nonzero squares and we will have two distinct representations for n^2. This is the case that gives motivation for this sequence.

LINKS

Table of n, a(n) for n=1..45.

EXAMPLE

50 is a term because 50 = 1^2 + 7^2 = 5^2 + 5^2 and 2500 = 14^2 + 48^2 = 30^2 + 40^2.

PROG

(PARI) isA273293(n) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == 2; }

lista(nn) = for(n=1, nn, if(isA273293(n) && isA273293(n^2), print1(n, ", ")));

CROSSREFS

Cf. A025285.

Sequence in context: A031692 A173141 A115592 * A097371 A179755 A186843

Adjacent sequences:  A273290 A273291 A273292 * A273294 A273295 A273296

KEYWORD

nonn

AUTHOR

Altug Alkan, May 19 2016

STATUS

approved

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Last modified August 25 13:50 EDT 2019. Contains 326324 sequences. (Running on oeis4.)