A240227 All even positive integers which have at least one 'balanced' representation as a sum of three distinct nonzero squares.

14, 26, 38, 42, 56, 62, 74, 78, 86, 98, 104, 114, 122, 126, 134, 146, 152, 158, 168, 182, 186, 194, 206, 218, 222, 224, 234, 248, 254, 258, 266, 278, 294, 296, 302, 312, 314, 326, 338, 342, 344, 350, 362, 366, 378, 386, 392, 398, 402, 416, 422, 434, 438, 446, 456, 458, 474, 482, 488, 494, 504, 518, 536, 542, 546, 554, 558, 566, 582, 584

Offset: 1

Author

Wolfdieter Lang, May 02 2014

Keywords

• nonn
• easy

Comments

For the numbers with representations as a sum of three distinct nonzero squares see A004432. For their multiplicity see A025442.

Here only even numbers are considered, and a representation 2*m = a^2 + b^2 + c^2, a > b > c > 0 denoted by the triple (a,b,c), is called 'balanced' if a = b + c. E.g., 62 is represented by (6, 5, 1) and (7, 3, 2) but only (6, 5, 1) is balanced because 6 = 5 + 1.

The multiplicities are given in A240228.

These numbers a(n) play a role in the problem proposed in A236300: Find all numbers which are of the form (x + y + z)*(u^2 + v^2 + w^2)/2, x >= y >= z >= 0, where u = x-y, v = x-z, w = y-z, with u >= 0, v >=0, w >= 0, u - v + w = 0 and even u^2 + v^2 + w^2 >= 4. The special case (called in a comment on A236300 case (iib)) with distinct u, v, and w, each >=1, needs the numbers a(n) of the present sequence. If the triple is taken as (u, u+w, w) with u < w then the [x, y, z] values are [2*u+w, u+w, u] and the number from A236300 is (2*u+w)*(u^2 + w^2 + u*w) =(2*u+w)*a(n). If this number from A236300 has multiplicity A240228(n) >=2 then there are A240228(n) different values for [x, y, z] and corresponding different A236300 numbers.

Examples

			n  a(n) (u, v=u+w, w)  [x, y,z]  A236300 member
1:  14   (1, 3, 2)    [4, 3, 1]     8*7 =   56
2:  26   (1, 4, 3)    [5, 4, 1]   10*13 =  130
3:  38   (2, 5, 3)    [7, 5, 2]   14*19 =  266
4:  42   (1, 5, 4)    [6, 5, 1]   12*21 =  252
5:  56   (2, 6, 4)    [8, 6, 2]   16*28 =  448
6:  62   (1, 6, 5)    [7, 6, 1]   14*31 =  434
7:  74   (3, 7, 4)   [10, 7, 3]   20*37 =  740
8:  78   (2, 7, 5)    [9, 7, 2]   18*39 =  702
9:  86   (1, 7, 6)    [8, 7, 1]   16*43 =  688
10: 98   (3, 8, 5)   [11, 8, 3]   22*49 = 1078 ...
For n=11 .. 20 see the link.
		

Links

• Wolfdieter Lang, The first twenty representations.

Crossrefs

Cf. A004432, A025442, A236300, A240228 (multiplicities).

Formula

The increasingly ordered elements of the set {2*k, k positive integer : 2*k = u^2 + (u+w)^2 + w^2 with 1 <= u < w }.

a(n) = 2*A024606(n). - Robert Israel, May 21 2014