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A215365 Primitive integer length of the side of an origin-centered square that contains inside its boundary a point with integer coordinates that is an integer distance from three of the four corners. 1
52, 700, 740, 996, 3364, 6240, 7800, 8400, 10952, 11184, 11352, 11492, 11484, 13156, 20280, 20988, 21320, 22472, 26180, 26588, 28168, 34500, 39988, 40680, 43700, 44944, 45976, 49500, 58956, 70448, 77500, 90168, 103896, 105468, 106200, 115752, 118636, 124620, 129000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

No point with integer distance to all four corners is known.

The sequence only contains even values because an odd-sided square centered at the origin has corners with non-integer coordinates, which cannot be at integer distance from interior lattice points. If the square instead of being centered at the origin has a corner on the origin, then the resulting sequence is A260549. - Giovanni Resta, Jul 29 2015

LINKS

Table of n, a(n) for n=1..39.

Yasushi Ieno, Other special cases of a square problem, arXiv:2111.02888 [math.GM], 2021.

Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.

G. Shute and K. L. Yocom, Problem 966. Seven integral distances, Math. Mag. 50, 166 (1977).

UnsolvedProblems Web Site, Rational Distance

EXAMPLE

With n = side length, we find an a,b such that a^2 + b^2 = d1^2, a^2 + (n-b)^2 = d2^2, b^2 + (n-a)^2 = d3^2, (n-a)^2 + (n-b)^2 = d4^2 is true in integers for three of these four equations. n = 52 is the first, with a=7 and b=24.

PROG

(PARI) has(n)=for(a=1, n-1, for(b=a, n-1, if(issquare(norml2([a, b])) + issquare(norml2([n-a, b])) + issquare(norml2([a, n-b])) + issquare(norml2([n-a, n-b])) > 2, return(1)))); 0

is(n)=sumdiv(n, d, has(d))==1 \\ Charles R Greathouse IV, Jul 28 2015

CROSSREFS

Cf. A260549.

Sequence in context: A254137 A249712 A255945 * A339142 A264309 A160344

Adjacent sequences:  A215362 A215363 A215364 * A215366 A215367 A215368

KEYWORD

nonn

AUTHOR

Mark Underwood, Aug 08 2012

EXTENSIONS

Data corrected and name improved by Mark Underwood, Jul 28 2015

a(7)-a(39) from Giovanni Resta, Jul 29 2015

STATUS

approved

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Last modified November 28 19:19 EST 2021. Contains 349415 sequences. (Running on oeis4.)