A173561 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).

1, 3, 7, 38, 47, 57, 157, 239, 829, 882, 993, 1772, 2673, 2917, 2943, 4747, 4952, 5257, 6118, 9466, 12943, 17557, 18543, 34208, 44179, 72662, 85353, 114669, 219602, 260359, 320078, 330182, 478707, 485298, 1083493, 1143007, 1477034, 1528649, 1615463, 1635786, 1984933

Offset: 1

Author

M. J. Knight (melknightdr(AT)verizon.net), Feb 21 2010

Keywords

• nonn

Comments

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.

Examples

			a(3) = 7 because 7^2+1 = 2*5^2 and 5/7 is smaller than all previous results.
		

Links

• Jonathan Bober, Dan Fretwell, Greg Martin, and Trevor D. Wooley, Smooth values of polynomials, Journal of the Australian Mathematical Society, Vol. 108, No. 2 (2020), pp. 245-261. arXiv:1710.01970 [math.NT] [alternate link]

Crossrefs

Cf. A006530, A014442.

Programs

Mathematica

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 *)

Extensions

More terms from Amiram Eldar, Mar 03 2021