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A358444
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a(1) = 1, a(2) = 2; for n > 2, a(n) = smallest positive number which has not appeared that has a common factor with a(n-2)^2 + a(n-1)^2.
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1
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1, 2, 5, 29, 4, 857, 10, 734549, 539562233501, 6, 12433, 15, 8, 17, 353, 12, 124753, 13, 14, 20, 16, 18, 22, 24, 25, 1201, 26, 41, 2357, 28, 5556233, 37, 30, 2269, 39, 32, 35, 52, 3929, 40, 15438641, 82, 45, 65, 34, 5381, 78, 50, 36, 38, 42, 44, 46, 48, 51, 3, 9, 21, 27, 33, 54, 55, 91
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OFFSET
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1,2
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COMMENTS
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The majority of terms are concentrated along or just above the line a(n) = n, resulting in 51 fixed points in the first 5000 terms. However, some terms are much larger because the sum of the squares of the previous two terms is a prime number.
Conjecture: the sequence is a permutation of the positive integers.
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LINKS
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Michael De Vlieger, Log-log scatterplot of a(n), n = 1..24857. a(24858) is a multiple of a prime factor of 2345424289569907866042152579118178340801^2 + 24922^2.
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EXAMPLE
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a(5) = 4 as a(3)^2 + a(4)^2 = 25 + 841 = 866, and 4 is the smallest unused number that shares a factor with 866.
a(9) = 539562233501 as a(7)^2 + a(8)^2 = 100 + 539562233401 = 539562233501, which is a prime number.
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MATHEMATICA
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nn = 120; c[_] = False; q[_] = 1; Do[Set[{a[i], c[i], q[i]}, {i, True, 2}], {i, 2}]; i = a[1]^2; j = a[2]^2; Do[k = i + j; s = FactorInteger[k][[All, 1]]; Do[(m = q[#]; While[c[# m], m++]; q[#] = m; If[# m < k, k = # m]) &[s[[n]]], {n, Length[s]}]; Set[{a[n], c[k], i, j}, {k, True, j, k^2}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 17 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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