OFFSET
1,1
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
LINKS
Eugen J. Ionascu, Ehrhart polynomial for lattice squares, cubes and hypercubes, arXiv:1508.03643 [math.NT], 2015.
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}.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 05 2013
STATUS
approved