A224448 Nonnegative numbers that have a representation as a sum of three primitive and distinct squares (square 0 allowed).

5, 10, 13, 14, 17, 21, 25, 26, 29, 30, 34, 35, 37, 38, 41, 42, 45, 46, 49, 50, 53, 54, 58, 59, 61, 62, 65, 66, 69, 70, 73, 74, 75, 77, 78, 81, 82, 83, 85, 86, 89, 90, 91, 93, 94, 97, 98, 101, 105, 106, 107, 109, 110, 113, 114, 115, 117, 118, 121, 122, 125, 126, 129, 130

Offset: 1

Author

Wolfdieter Lang, Apr 09 2013

Keywords

• nonn

Comments

These are the numbers a(n) satisfying A224447(a(n)) = k >= 1, and k gives their multiplicity. See the comments on A224447 for more details and a F. Halter-Koch corollary (Korollar 1. (c), p. 13 with the first line of r_3(n) on p. 11) according to which this sequence gives the increasingly ordered numbers satisfying: neither congruent 0, 4, 7 (mod 8) nor a member 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, ?}, with a number $ >= 5*10^10 if it exists at all.

Examples

			Denote a representation in question by the triple [a, b, c].
The representations for n= 1, 2, ..., 10 are:
n=1,   5: [0, 1, 2],
n=2,  10: [0, 1, 3],
n=3,  13: [0, 2, 3],
n=4,  14: [1, 2, 3],
n=5,  17: [0, 1, 4], [2, 2, 3],
n=6,  21: [1, 2, 4],
n=7,  25: [0, 0, 5], [0, 3, 4],
n=8,  26: [0, 1, 5], [1, 3, 4],
n=9,  29: [0, 2, 5], [2, 3, 4]
n=10, 30: [1, 2, 5].
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.

Crossrefs

Cf. A224447, A047449 (primitive case).

Programs

Mathematica

representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[130], representableQ] (* Jean-François Alcover, Apr 10 2013 *)

Formula

a(n) is the n-th largest number m satisfying m = a^2 + b^2 + c^2, with a, b, and c integers, 0 <= a < b < c, and gcd(a,b,c) = 1.

a(n) is the n-th largest number m for which A224447(m) > 0.