A196943 -id:A196943 - cp's OEIS frontend

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

Christopher Monckton of Brenchley, Oct 08 2011

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

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

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.

Robin Visser, Table of n, a(n) for n = 1..10000
E. W. Weisstein, MathWorld: Euler brick

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

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

A306120 Lengths of largest face diagonal in primitive Euler bricks or Pythagorean cuboids: possible values of max(d, e, f) for solutions to a^2 + b^2 = d^2, a^2 + c^2 = e^2, b^2 + c^2 = f^2 in coprime positive integers a, b, c, d, e, f.

Original entry on oeis.org

267, 373, 732, 825, 843, 1595, 1884, 2500, 2775, 3725, 3883, 6380, 6409, 8140, 8579, 9188, 9272, 9512, 11764, 12125, 13123, 14547, 14681, 14701, 19572, 20503, 20652, 24695, 25121, 25724, 29307, 30032, 30695, 31080, 32595, 34484, 37104, 37895, 38201, 38965

Offset: 1

M. F. Hasler, Oct 11 2018

nonn

These are the values obtained as sqrt(A031173(n)^2 + A031174(n)^2), sorted by size (n = 3 yields 843, n = 4 yields 732) and duplicates removed: The first duplicate is 71402500^2 = A031173(1428)^2 + A031174(1428)^2 = A031173(1626)^2 + A031174(1626)^2, there is no other among the first 3500 terms.
This considers only the face diagonals, not the space diagonals.
See the main entry A031173 for links, cross-references, and further comments.

M. F. Hasler, Table of n, a(n) for n = 1..2500

Cf. A031173, A031174, A031175, A195816, A196943, A268396.

A306120=Set(vector(1000,n,sqrtint(A031173(n)^2+A031174(n)^2)))[1..-100] \\ Discard the last 100 values, which may have holes. This is empirical: better find the smallest sqrtint(A031173(n)^2+A031174(n)^2) with n > 1000 not in the set, and discard all elements larger than that.

A306120 = { sqrtint(A031173(n)^2+A031174(n)^2); n >= 1 }.

cp's OEIS Frontend

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

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