OFFSET
1,1
COMMENTS
These are the increasingly ordered numbers a(n) for which A223730(a(n)) = 3. See also A223731. These are the numbers n with exactly three 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(185) = 4075 = 31^2 + 33^2 + 45^2 = 23^2 + 39^2 + 45^2 = 5^2 + 9^2 + 63^2 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..185
FORMULA
This sequence lists the increasingly ordered distinct members of the set S3:= {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, and there are exactly three different solutions for this m}.
EXAMPLE
a(1) = 66 because the smallest number n with A223730(n) = 3 is 66. The three solutions for m = 66 are denoted by [1,1,8], [1,4,7] and [4,5,5].
For n=2..10 the three representations of a(n) are given by
n=2, 86: [1, 2, 9], [1, 6, 7], [5, 5, 6],
n=3, 89: [2, 2, 9], [2, 6, 7], [3, 4, 8],
n=4, 101: [1, 6, 8], [2, 4, 9], [4, 6, 7],
n=5, 110: [1, 3, 10], [2, 5, 9], [5, 6, 7],
n=6, 114: [1, 7, 8], [4, 7, 7], [5, 5, 8],
n=7, 131: [1, 3, 11], [1, 7, 9], [5, 5, 9],
n=8, 149: [1, 2, 12], [2, 8, 9], [6, 7, 8],
n=9, 153: [2, 7, 10], [4, 4, 11], [5, 8, 8],
n=10: 166: [2, 9, 9], [3, 6, 11], [6, 7, 9].
For n = 153 there is also the non-primitive representation [6,6,9] = 3*[2,2,3] not counted here.
MATHEMATICA
threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[500], threeSquaresCount[#] == 3 &] (* Jean-François Alcover, Jun 21 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 05 2013
STATUS
approved