A355901 Records in A355899.
1, 3, 5, 11, 17, 71, 579, 2705
Offset: 1
Programs
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Mathematica
nn = 2^16; j = k = r = 1; {1}~Join~Reap[Do[m = # + (j + k)/# &@ GCD[j, k]; If[# > r, r = #; Sow[r]] &[GCD[k, m]]; j = k; k = m, {n, 3, nn}]][[-1, -1]] (* Michael De Vlieger, Sep 03 2022 *)