A223732 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly one way.

3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 27, 29, 30, 34, 35, 42, 43, 45, 46, 49, 50, 53, 61, 65, 67, 70, 73, 75, 78, 82, 91, 93, 97, 106, 109, 115, 133, 142, 145, 147, 157, 163, 169, 190, 193, 202, 205, 235, 253, 265, 277, 298, 397, 403, 427, 442, 445, 505, 793

Offset: 1

Author

Wolfdieter Lang, Apr 05 2013

Keywords

• nonn

Comments

These are the increasingly ordered numbers a(n) for which A233730(a(n)) = 1. See also A233731. These are the numbers n with exactly one 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: 793 = 6^2 + 9^2 + 26^2 is the largest element of this sequence. - Alois P. Heinz, Apr 06 2013

Examples

			a(1) = 3 because there is no solution for m = 1 and 2 as a primitive sum of three nonzero squares, and m = 3 = 1^2 + 1^2 + 1^2 is the only solution with [a,b,c] = [1,1,1].
a(5) = 14 because 14 is the fifth largest member of the set S1, and [a,b,c] = [1,2,3] denotes this unique representation for m = 14.
		

Links

• Eugen J. Ionascu, Ehrhart polynomial for lattice squares, cubes and hypercubes, arXiv:1508.03643 [math.NT], 2015.

Crossrefs

Cf. A233730, A233731, A233733, A233734.

Programs

Mathematica

threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[800], threeSquaresCount[#] == 1 &] (* Jean-François Alcover, Jun 21 2013 *)

Formula

This sequence lists the increasingly ordered members of the set S1 := {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, gcd(a,b,c) = 1, with only one such solution for this m}.