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A197040
Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.
1
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
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
KEYWORD
nonn,base
AUTHOR
STATUS
approved