A224447 -id:A224447 - cp's OEIS frontend

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

Wolfdieter Lang, Apr 09 2013

nonn

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.

			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].

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.

Cf. A224447, A047449 (primitive case).

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

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.

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

Original entry on oeis.org

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

Wolfdieter Lang, Apr 09 2013

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.

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

Cf. A224447, A224448, A051952.

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 *)

cp's OEIS Frontend

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

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