A224449 A finite set of numbers relevant for the representation of numbers as primitive distinct sums of three squares (0 squared allowed).

1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627

Offset

1,2

Comments

This set of 23 numbers, possibly with one more number a >= 5*10^10, appears in a corollary of the Halter-Koch reference (Korollar 1.(c), p. 13 with the first line of r_3(n) on p. 11). A number is representable as a^2 + b^2 + c^2 with a,b, and c integers, 0 <= a < b < c, and gcd(a,b,c) = 1 if and only if n is not congruent 0, 4, 7 (mod 8) and not one of the numbers {a(k), k = 1 .. 23}, and, if it exists at all, a further number >= 5*10^10.

For the multiplicities of these representable numbers see A224447, and for the numbers themselves see A224448.

For a similar set of numbers relevant for sums of three nonzero squares see A051952.

Links

Table of n, a(n) for n=1..23.

F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.

Mathematica

representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[1000], Not[ representableQ[#] || MatchQ[ Mod[#, 8], 0 | 4 | 7]] &] (* Jean-François Alcover, Apr 10 2013 *)

Crossrefs

Cf. A224447, A224448, A051952.

Sequence in context: A358130 A176189 A057127 * A007780 A018537 A018331

Adjacent sequences: A224446 A224447 A224448 * A224450 A224451 A224452

Keyword

nonn,fini

Author

Wolfdieter Lang, Apr 09 2013

Status

approved