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A240227 All even positive integers which have at least one 'balanced' representation as a sum of three distinct nonzero squares. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..70.

Wolfdieter Lang, The first twenty representations.

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

EXAMPLE

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.

CROSSREFS

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

Sequence in context: A235688 A176274 A344872 * A191992 A082773 A242395

Adjacent sequences:  A240224 A240225 A240226 * A240228 A240229 A240230

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 02 2014

STATUS

approved

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Last modified June 25 04:52 EDT 2021. Contains 345452 sequences. (Running on oeis4.)