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%I #6 Jul 25 2021 10:18:35
%S 1,3,7,38,47,57,157,239,829,882,993,1772,2673,2917,2943,4747,4952,
%T 5257,6118,9466,12943,17557,18543,34208,44179,72662,85353,114669,
%U 219602,260359,320078,330182,478707,485298,1083493,1143007,1477034,1528649,1615463,1635786,1984933
%N Numbers k such that gpf(k^2+1)/k sets a new record of low value, where gpf(k) is the greatest prime dividing k (A006530).
%C This is an infinite sequence, since the solutions to the Pell equations for primes p = 4*k+1 will give ratios with limit 0. For example, the entry 7 satisfies 7^2 - 2*5^2 = -1 and the ratio is 5/7. However, not all entries are given by this technique.
%H Jonathan Bober, Dan Fretwell, Greg Martin, and Trevor D. Wooley, <a href="https://arxiv.org/abs/1710.01970">Smooth values of polynomials</a>, Journal of the Australian Mathematical Society, Vol. 108, No. 2 (2020), pp. 245-261. arXiv:1710.01970 [math.NT] [<a href="https://www.sunsite.ubc.ca/~gerg/papers/downloads/SVP.pdf">alternate link</a>]
%e a(3) = 7 because 7^2+1 = 2*5^2 and 5/7 is smaller than all previous results.
%t f[n_] := FactorInteger[n^2 + 1][[-1, 1]]/n; s = {}; fm = 3; Do[f1 = f[n]; If[f1 < fm, fm = f1; AppendTo[s, n]], {n, 1, 2*10^4}]; s (* _Amiram Eldar_, Mar 03 2021 *)
%Y Cf. A006530, A014442.
%K nonn
%O 1,2
%A M. J. Knight (melknightdr(AT)verizon.net), Feb 21 2010
%E More terms from _Amiram Eldar_, Mar 03 2021
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