OFFSET
1,1
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.
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.
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.
EXAMPLE
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].
MATHEMATICA
representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[130], representableQ] (* Jean-François Alcover, Apr 10 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 09 2013
STATUS
approved