OFFSET
0,27
COMMENTS
a(n), n >= 1 gives the number of different representations of the positive integer n as a sum of three distinct squares (square 0 allowed) which have no common factor > 1. a(0) = 0. Neither the order of the summands nor the signs of the numbers to be squared are taken into account. If a(n) = 0 there is no such representation for n.
According to a corollary by F. Halter-Koch (Korollar 1. (c), p. 13, together with the first line of r_3(n) on p. 11) a(n) > 0 if and only if n is neither congruent 0, 4, 7 (mod 8) nor an element of the set S := {1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627, ?}, and the number ? >= 5*10^10 if it exists at all. This set appears as A224449.
See A224444 for the multiplicities for primitive sums of three squares (square 0 allowed).
The numbers for which a(n) is not 0 are given in A224448.
LINKS
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.
FORMULA
a(n) = k >= 1 if n, n >= 0, has k different representations as n = a^2 + b^2 + c^2, a, b and c integers, 0 <= a < b < c and gcd(a,b,c) = 1. If there is no such representation a(0) = 0.
EXAMPLE
Denote a representation in question by an increasingly ordered triple [a, b, c].
The first nonnegative integer with a representation in question is n = 5 with a(5) = 1 because 5 has only one primitive representation (see A224444(5) = 1), namely [0, 1, 2] and the entries are distinct.
a(6) = 0 because the only primitive representation (A224444(6) = 1) is [1, 1, 2], but the entries are not distinct.
a(17) = 1 with the unique representation [0, 1, 4]. The primitive representation [2, 2, 3] (A224444(17) = 2) is excluded because it does not have distinct entries.
a(26) = 2 with the primitive representations (A224444(26) = 2) given by [0, 1, 5] and [1, 3, 4] which both have distinct entries.
MATHEMATICA
a[n_] := Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 &] // Length; Table[a[n], {n, 0, 130}] (* Jean-François Alcover, Apr 10 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 09 2013
STATUS
approved