A197040 Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.

3, 8, 9, 8, 9, 9, 6, 9, 10, 8, 7, 9, 6, 8, 7, 8, 11, 6, 7, 8, 9, 8, 7, 6, 8, 10, 6, 6, 6, 8, 8, 8, 8, 9, 6, 9, 7, 6, 7, 8, 8, 9, 7, 11, 7, 8, 5, 9, 8, 9, 9, 7, 6, 7, 9, 6, 7, 9, 7, 8, 10, 5, 9, 7, 7, 7, 7, 6, 9, 9, 6, 8, 7, 9, 8, 6, 9, 5, 9, 9, 8, 6, 6, 7, 7

Offset

1,1

Comments

Distribution of edge-length occurrences for Euler bricks is remarkably near-uniform.

References

L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.

P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.

Links

Robin Visser, Table of n, a(n) for n = 1..10000

E. W. Weisstein, MathWorld: Euler brick

Example

For n=1 (i.e., the integers 1..100), there are only 3 possible edge-lengths for Euler bricks: 44, 85, 88.

Prog

(Sage)
def a(n):
    ans = set()
    for x in range(100*(n-1)+1, 100*n+1):
        divs = Integer(x^2).divisors()
        for d in divs:
            if (d <= x^2/d): continue
            if (d-x^2/d)%2==0:
                y = (d-x^2/d)/2
                for e in divs:
                    if (e <= x^2/e): continue
                    if (e-x^2/e)%2==0:
                        z = (e-x^2/e)/2
                        if (y^2+z^2).is_square(): ans.add(x)
    return len(ans)  # Robin Visser, Jan 02 2024

Crossrefs

cf. A195816, A196943, A031173, A031174, A031175. Edge lengths of Euler bricks are A195816; face diagonals are A196943.

Sequence in context: A179047 A185067 A256955 * A217870 A154927 A348297

Adjacent sequences: A197037 A197038 A197039 * A197041 A197042 A197043

Keyword

nonn,base

Author

Christopher Monckton of Brenchley, Oct 08 2011

Status

approved