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A308834 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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
This sequence was suggested to me by Moshe Shmuel Newman.
This is A135998 with the exponent 4 replacing 3.
From M. F. Hasler, Feb 03 2024: (Start)
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)
LINKS
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A135998 (equivalent for 3rd powers).
Sequence in context: A043163 A043943 A367360 * A084049 A251221 A302467
KEYWORD
sign
AUTHOR
David S. Newman, Jun 27 2019
STATUS
approved

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Last modified April 13 17:26 EDT 2024. Contains 371644 sequences. (Running on oeis4.)