A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

9, 17, 18, 25, 26, 27, 29, 33, 34, 36, 38, 45, 49, 51, 53, 57, 59, 61, 62, 68, 69, 72, 73, 75, 77, 82, 83, 85, 94, 97, 100, 102, 104, 105, 106, 107, 108, 109, 116, 118, 123, 130, 132, 136, 138, 139, 141, 144, 147, 152, 154, 155, 157, 158, 165, 177, 180, 187

Offset: 1

Author

Wolfdieter Lang, Apr 08 2013

Keywords

• nonn

Comments

These are the numbers for which A000164(a(n)) = 2.

a(n) is the n-th largest number which has a representation as sum of three integer squares (square 0 allowed), in exactly two ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity for a(n) with order and signs taken into account is A005875(a(n)).

This sequence is a proper subsequence of A000378.

Examples

			a(1) = 9 = 0^2 + 0^2 + 3^2 = 1^2 + 2^2 + 2^2, and 9 is the smallest number m with A000164(m) = 2 for m >= 0. The multiplicity with order and signs taken into account is 2*3 + 8*3 = 30 = A005875(9).
The two representations [a,b,c] for a(n), n = 1, ..., 10, are
n=1,   9 = [0, 0, 3], [1, 2, 2],
n=2,  17 = [0, 1, 4], [2, 2, 3],
n=3,  18 = [0, 3, 3], [1, 1, 4],
n=4,  25 = [0, 0, 5], [0, 3, 4],
n=5,  26 = [0, 1, 5], [1, 3, 4],
n=6,  27 = [1, 1, 5], [3, 3, 3],
n=7,  29 = [0, 2, 5], [2, 3, 4],
n=8,  33 = [1, 4, 4], [2, 2, 5],
n=9,  34 = [0, 3, 5], [3, 3, 4],
n=10, 36 = [0, 0, 6], [2, 4, 4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000164, A005875, A000378, A094942 (one way), A224443 (three ways).

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^22, 3, min(3, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
    end:
a:= proc(n) option remember; local k;
      for k from 1 +`if`(n=1, 0, a(n-1))
      while b(k, isqrt(k), 3)<>2 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 09 2013

Mathematica

Select[ Range[0, 200], Length[ PowersRepresentations[#, 3, 2]] == 2 &] (* Jean-François Alcover, Apr 09 2013 *)

Formula

This sequence gives the increasingly ordered elements of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly two such representation}.

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 2, m >= 0}.