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A223733
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Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly two ways.
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5
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33, 38, 41, 51, 54, 57, 59, 62, 69, 74, 77, 81, 83, 90, 94, 98, 99, 102, 105, 107, 113, 117, 118, 121, 122, 123, 125, 126, 137, 138, 139, 141, 150, 154, 155, 158, 162, 165, 170, 177, 178, 181, 187, 195, 197, 203, 210, 211, 213, 214, 217, 218, 225, 226, 229
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OFFSET
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1,1
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COMMENTS
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These are the increasingly ordered numbers a(n) for which A223730(a(n)) = 2. See also A223731. These are the numbers n with exactly two representation as a primitive sum of three nonzero squares (not taking into account the order of the three terms, and the number to be squared for each term is taken positive).
Conjecture: a(147) = 1885 = 16^2 + 27^2 + 30^2 = 12^2 + 29^2 + 30^2 is the largest element of this sequence. - Alois P. Heinz, Apr 06 2013
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LINKS
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FORMULA
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This sequence lists the increasingly ordered distinct members of the set S2:= {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, and there are exactly two different solutions for this m}.
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EXAMPLE
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a(1) = 33 because the smallest number n with A223730(n) = 2 is 33. The two representations of 33 are denoted by [1, 4, 4], and [2, 2, 5].
The two representations for a(n) for n = 2..10 are denoted by
n=2, 38: [1, 1, 6], [2, 3, 5],
n=3, 41: [1, 2, 6], [3, 4, 4],
n=4, 51: [1, 1, 7], [1, 5, 5],
n=4, 54: [1, 2, 7], [2, 5, 5], ([3, 3, 6] is non-primitive)
n=5, 57: [2, 2, 7], [4, 4, 5],
n=6, 59: [1, 3, 7], [3, 5, 5],
n=7, 62: [1, 5, 6], [2, 3, 7],
n=8, 69: [1, 2, 8], [2, 4, 7],
n=9, 74: [1, 3, 8], [3, 4, 7],
n=10, 77: [2, 3, 8], [4, 5, 6].
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MATHEMATICA
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threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[300], threeSquaresCount[#] == 2 &] (* Jean-François Alcover, Jun 21 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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