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A308834
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a(n) is the smallest error in trying to solve n^4 = x^4 + y^4. That is, for each n from 2 on, find positive integers x and y, x <= y < n such that |n^4 - x^4 - y^4| is minimal and let a(n) = n^4 - x^4 - y^4.
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4
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14, 49, 94, 113, 46, -191, 399, 64, -657, 545, -466, -721, -145, 1328, 270, -2751, 719, -751, 1118, -1376, -1041, 1839, 1310, 1663, 815, 5184, -306, 9104, 863, 1455, 4320, 7024, -5105, 4289, 11504, 64, -12016, 2816, 10799, -11200, 6094, -2671, -226, 20753
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OFFSET
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2,1
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COMMENTS
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This is A135998 with the exponent 4 replacing 3.
Can it happen that (x,y) and (x',y') yield the same minimal absolute difference, but with opposite signs? If so, how is a(n) defined in this case?
Without the condition y < n, the trivial "solution" (x, y) = (1, n) would always yield a(n) = -1. With the condition, there is no admissible pair (x,y) for n = 1, whence a(1) is undefined. (End)
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LINKS
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EXAMPLE
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Here are the calculations for the first few values.
For 2, the only possible values for x and y are 1,1, so we have
a(2) = 2^4 - 1^4 - 1^4 = 16 - 2 = 14.
For 3, y can be 1 or 2. if y is 1, x is 1 as well, and if y=2, then x can be 1 or 2.
3^4 - 1^4 - 1^4 = 79
3^4 - 1^4 - 2^4 = 64
3^4 - 2^4 - 2^4 = 49.
The smallest absolute value is in the last case, so a(3) = 49.
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MATHEMATICA
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nend = 100; For[n = 2, n <= nend, n++, a[n] = 0]; For[n = 2, n <= nend, n++, min = n^4; For[y = 1, y <= n - 1, y++, For [x = y, x <= n - 1, x++, changed = False; sol = n^4 - x^4 - y^4; If[(sol < min) && (sol > 0), min = sol; changed = True]; If[(Abs[sol] < min) && (sol < 0), min = -sol; changed = True]; If[changed, a[n] = sol]]]]; Print[t = Table[a[i], {i, 2, nend}]] (* or *)
a[n_] := SortBy[n^4 - Flatten[Table[x^4 + y^4, {x, n-1}, {y, x}]], Abs][[1]]; Array[a, 99, 2] (* Giovanni Resta, Jul 05 2019 *)
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PROG
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(PARI) A308834(n, p=4) = { my(np=n^p, m=np); for(y=max(sqrtnint(np\2, p), 1), n-1, my(x = sqrtnint(np - y^p, p), dy = np-y^p, d = if(dy-x^p > (x+1)^p-dy && x < n-1, dy-(x+1)^p, dy-x^p)); abs(d) < abs(m) && abs(m=d) < 2 && break); m} \\ M. F. Hasler, Feb 03 2024
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CROSSREFS
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Cf. A135998 (equivalent for 3rd powers).
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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