A223734 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly three ways.

66, 86, 89, 101, 110, 114, 131, 149, 153, 166, 171, 173, 174, 179, 182, 185, 186, 189, 198, 219, 221, 222, 227, 233, 234, 237, 241, 242, 245, 258, 261, 270, 274, 286, 291, 294, 302, 305, 309, 318, 323, 334, 338, 347, 349, 361, 363, 366, 377, 378, 387, 405, 410

Offset: 1

Author

Wolfdieter Lang, Apr 05 2013

Keywords

• nonn

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..185

Crossrefs

Cf. A223730, A223731, A223732, A223733.

Programs

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 *)

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}.