OFFSET
1,1
COMMENTS
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..147
FORMULA
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}.
EXAMPLE
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].
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 05 2013
STATUS
approved