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A240227
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All even positive integers which have at least one 'balanced' representation as a sum of three distinct nonzero squares.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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 }.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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