A108399 Least positive k such that n^2 + k is a golden semiprime (A108540).

5, 2, 6, 61, 52, 41, 28, 13, 106, 87, 66, 43, 18, 393, 364, 333, 300, 265, 228, 189, 148, 105, 60, 13, 226, 175, 122, 67, 10, 463, 402, 339, 274, 207, 138, 67, 814, 739, 662, 583, 502, 419, 334, 247, 158, 67, 538, 443, 346, 247, 146, 43, 4494, 4387, 4278, 4167, 4054

Offset: 1

Author

Jason Earls, Jul 03 2005

Keywords

• easy
• nonn

Comments

Conjecture: for every n > 1 there exists a number k < n^3 such that n^2 + k is a golden semiprime.

Examples

			a(4)=61 because 4^2+61 = 77 = 7*11 and 7*phi-11 = 0.326237... < 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A108540.

Programs

Mathematica

goldQ[n_] := Module[{f = FactorInteger[n]}, If[Length[f] != 2, False, If[Max[f[[;;,2]]] != 1, False, Abs[f[[2,1]] - f[[1,1]] * GoldenRatio] < 1]]]; a[n_] := Module[{k = 1}, While[!goldQ[n^2 + k], k++]; k]; Array[a, 57] (* Amiram Eldar, Nov 29 2019 *)